Overlayin the Just-in-Time with Kanban System on an American

Production Environment

by

Patrick Robert Philipoom

Dis ertation submitted to the Faculty of the

Virqini Polytechnic Institute and State University

in partial ulfillment of the requirements for the degree of

Doctor of Philosophy

in

Management Science

APPROVED:

L. P. Rees, Chairman

/

a:. . Moore B. W. T~@~, III

February, 1986

Blacksburg, Virginia

Overlaying the Just-in-Time with Kanban System on an American Pro-

duction Environment

by

Patrick Robert Philipoom

L. P. Rees, Chairman

Management Science

(ABSTRACT)

During the past several years, the publicized successes

of Japanese production management techniques have created an

interest in tJe potential of these techniques for application

in an Americaf manufacturing environment. One such Japanese

technique thal has been the focus of much attention from

American manufacturers and production managers is the "just-

in-time ( JIT) I" technique implemented with "Kanbans. " 1 How-

ever, the ap+ications of the JIT technique in Japan that

have been rep0rted have been for large scale assembly line

operations thrt, in general, encompass the unique physical

and philosophii.cal characteristics typical of Japanese pro-. . I

duction systems.

The factor~ that contribute to the success of the JIT

system in Jap In are frequently not exhibited in manufacturing

systems in th United States, especially in American systems

1 Toyota uses a system o]f cards, called Kanbans, to control inventory and schedu~e production in their automotive assembly plants. I

i I I I

_that combine as.sembly and shop~type operations and encompass

a high degree of system v.ariabili ty. As such, it is ques-

tionable whether the JIT technique can be successfully

adapted to American manufacturing systems~that do not display

the characteristics of Japanese production operations. Nev-

ertheless, a number of American manufacturing companies, in

hope of achieving at least some of the Japanese success in

inventory control, quality control and production scheduling,

have begun implementing the JIT technique in their own unique

production environment. The purpose of this dissertation is

to investigate implementing JIT in a non-Japanese production

environment and to show how JIT can be adapted so that it can

have a broader range of applicability, especially under the

particular set of conditions that are very likely to exist

in many American production environments.

ACKNOWLEDGEMENTS

I would like to thank Dr. Loren P. Rees, my chairman and

mentor, for the endless support, effort and guidance that he

has provided throughout my graduate studies at Virginia Tech

especially during the dissertation stage of my doctoral

program.

I would like to thank Dr. Bernard W. Taylor, III, Head of

the Department of Management Science, for his provision of

financial aid and resources during my advanced studies and

especially for his help in developing three journal articles

from the chapters of this dissertation.

I would like to thank Dr. Phi lip Y. Huang for making

available his expert insight in Just-in-Time production sys-

terns.

I would like to thank Dr. Edward R. Clayton and Dr.

Laurence J. Moore for their assistance with the dissertation

and their support and encouragement throughout my graduate

studies at Virginia Tech.

I would like to thank Dr. Roberta S. Russell for kindly

agreeing to attend the defense of my dissertation in place

of Dr. Huang.

I would like to thank

for their cheerful dispositions and their help _with the typ-

ing of this d~ssertation.

Acknowledgements !

iv

Finally, I would like to thank my parents,

f 9r their continual love and support

throughout this and a~l of my endeavors.

Acknowledgements v

TABLE OF CONTENTS

1 . O· Introduction ' 1. 1 Types of Producti1on Shops

1.1.1 Classificatio~s By.Production Volume

1. 1.1. 1 Job Shop Production

1.1.1.2 Batch Production I

1.1.1. 3 Mass Prodiuction !

1.1.2 Classificatio~s By Plant Layout

1.1.2.1 Fixed Pos~tion Layout I

1.1.2.2 Process L~yout

1.1. 2. 3 Product Filow Layout

1.2 Alternative Produ~tion Systems !

1.2.1 Order Point S~stems

1.2 .2 i .

Material Regu~rements Planning

1.3 Just-in-Time . . . . I

I

1.3.1 Cultural Infl~ences in Japan·

1.3.1.1 Respect fpr Humanity

1. 3 .1. 2 Lifetime :Emp~ __ ?yment

1. 3. 1. 3 Vendor Reilationships i

1.3.2 Assumptions Necessary for JIT Systems !

1. 3. 2 .1 Small Set}ip Times I I

1.3.2.2 Frozen De~and Schedule

1.3.2.3 WorkforceAttributes

1.3.2.4 Quality Control

Table of Contents

1

4

4

4

5

5

6

6

6

7

7

7

9

12

. . . 14

14

15

15

16

16

17

18

19

vi

,/ 1.4 Toyota's Kanban ystem 20

1. 5 Comparison of Kaban with MRP 23 "" 1. 6 Review of Quantitative Research in Kanban 26 ./ vl.-7 Motivation and S ope of Investigation 28 \/""

2.0 The Example Shop . . . . 31

2.1 The Product Structure and Example Shop 31

2.2 The Simulation M del 34

2.2.1 The Assembly Workcenter 35

2.2.2 The Job Workcenter 37

2.3 Summary 40

3.0 Determining Workcen er Lotsizes For Signal Kanbans 41 ./

3.1 The Just-in-Time System with Signal Kanbans 43

3. 2 Case Example and Simulation Model of a Shop Operation 49

3.3 An Integer Progrrmming Model for Determining

Lotsizes . . . . r . .

3.3.1 The Multi-Pr0duct EOQ Model

3.3.2 Model Constrlints ..... .

3.3.3 The Inventort Minimization Model

3.3.4 The Cost Minimization Model

3.4 Simulation Analytis Of Model Lotsizes

3.5 Simulation Analysis of Container Processing Times

3.5.1 Comparing Siinal and Standard Kanban Systems I

3.6 Summary

Table of Contents

53

53

56

60

65

66

69

73 v 74

vii

i . I i

!

I 4.0 An Analysis of Factol1s Influencing the Number of Kanbans Re-

quired at a Workcenter . . . . . . . . . • . . . 76 V" I

4.1 A Japanese Appro,ch to Determining the Number of

. . . . . I .

I Kanbans 76 ~

4. 2 Factors Influenci[ng The Number of Kanbans to Use in I

a JIT System With Va~iation 78

4. 2. 1 A Simulation lease Example 84 I

4. 3 A Simulation Appr1oach for Determining the Initial

Number of Kanbans at a Workcenter

4.3.1 The Simulation Model

4.3.2 The Effect of Less-than-Ideal Production Fae-

tors

:1 4.4 Summary

5.0 Dynamically Adjusting! the Number of Kanbans Using Estimated

Values of Leadtime . · 1 · . . . . . . . . . . . . . . . .

5.1 Methodology for Dlnamically Adjusting Kanbans

5.1.1 Special Case ff the Methodology: Shortage Costs

Overwhelm Holding cbsts

5.2 case Examples

5. 2 .1 Example 1: I lilustration of the Methodology

5.2.2 Example 2: Adbusting to Kanban Misspeci-

fications • : • i • • • •

I

3: A fraining 5.2.3 Example Case Example I

• I • • • •

I

5.3 Summary

!

Table of Contents

89

92

98

103

105

108

123

126

129

136

140

141

viii

6.0 Summary and Concfu. ions

6.1 Extensions and Future Research

Bibliography • • e • e • e e e e e I I I I I II I I ' I I I I

Vita I I I I I I I I I I e e 0 I I I I I I I I I I e 8 I I I II I

Table of Contents

145

152

154

160

ix

1.0 INTRODUCTION

During the past seyeral years, the publicized successes

of Japanese productio1 management techniques have created an

interest in the potential of these techniques for application

in an American rnanufaclturing environment. One such Japanese I

technique that has b~en the focus of much q.ttention from

American manufacturer, and production managers is the "just-

in-time (JIT)" techniqµe implemented with "Kanbans." However,

the applications of th~ JIT technique in Japan that have been

reported have been for1 large scale assembly line operations

that, in general, encompass the unique physical and phi lo-

sophical characteristics typical of Japanese production sys-

terns.

The factors that contribute to the success of the JIT

system in Japan are friquently not exhibited in manufacturing

systems in the United States, especially in American systems

that combine assembly and shop-type operations and encompass

a high degree of system variability. As such, it is ques-

tionable whether the JIT technique can be successfully

adapted to American manufacturing systems that do not display . I .

the characteristics of Japanese production operations. Nev-

ertheless, a number of American manufacturing companies, in I

hope of achieving at !least some of the Japanese success in I

inventory control, quall..ity control and production scheduling,

Introduction 1

have begun implementing the JIT technique in their own unique

production environment.

The purpose of this dissertation is to investigate imple-

menting JIT in a non-Japanese production environment and to

show how JIT can be adapted so that it will have a broader

range of applicability, especially under the particular set

of conditions that are very likely to exist in many American

production environments. The objectives are to ·determine:

1. how JIT can be used even when the firm cannot reduce the

setup times for all machines sufficiently to allow single

container lotsizes;

2. the factors that are present in an American production

environment that ·influence the number of Kanbans in a

shop, and how to set the initial number of Kanbans in

such an environment; and

3. how the number of Kanbans at a workcenter should be dy-

namically adjusted when conditions in the shop are not

stable.

Before the investigation conducted in this dissertation

can be presented~ certain terms and concepts must be classi-

fied and/or defined; this is done in the remainder of this

chapter. First, two classifications of production shops are

Introduction 2

given to help facilitate the discussion of how the sample

shop used in this investigation (which is described in Chap-

ter 2) differs from the shops in which JIT has been used by

the Japanese. Next, two alternative production systems, or-

der point and material requirements planning, are discussed.

The Kanban system is essentially an order point system,

whereas material requirements planning is a system widely

used in the United States for dependent demand.

Having ci.eveloped a framework as to different types of

shops and different types of systems, the philosophical and

cultural differences between Japanese and American production

systems are examined. These include such factors as respect

for humanity, lifetime employment, frozen demand schedules

and workforce attributes. Then, the JIT system developed by

Toyota, the JIT with Kanban system is described, and the role

that the Kanban system plays within the overall JIT system

is delineated. The JIT with Kanban system is also compared

to material requirements planning.

Finally, a review of the quantitative research in JIT is

presented. The current literature is summarized and its lim-

itations are discussed. This provides a focus for the moti-

vation and scope of the investigation which follows.

Introduction 3

1. 1 TYPES OF PRODUCTION SHOPS

There are many ways to classify production shops. Groover \

[ 13] suggests two schemes: by the production volume and by

the layout of the plant. Each type of shop under these clas-

sifications is briefly summarized and references are provided

for the interested reader.

1.1.1 CLASSIFICATIONS BY PRODUCTION VOLUME

Classification of production shops by production volume

is based on the fact that as the volume of production in-

creases, benefits accrue by using more specialized machines

and layouts.

1. 1. 1. 1 Job Shop Production

Job shop production [55, pp. 27-48] involves the manufac-

ture of small lots with low production volumes. Shops within

this category produce orders to meet specific custom.er re-

quests, which are often one-time orders. A job shop must have

general purpose production equipment and highly skilled

workers because of the variety of products it manufactures.

Such a shop is typically laid out to allow for great flexi-

bility in machining requirements.

Introduction 4

1. 1. 1.2 Batch Production

Batch production·[55, pp. 49-76] is used for manufacturing

medium-sized lots for orders that tend to be somewhat repet ...

itive and continuous. There is usually less variety than with

job shop production in the types of orders received. This

homogeneity allows the shop to use more specialized equip-

ment, which reduces much of the labor cost in producing an

i tern and enables the production rate to be increased. With

batch production, a batch is produced and placed in inven-

tory; when the inventory is depleted or reduced to low lev-

els, another batch is produced.

1. 1. 1 .3 Mass Production

This category of shop is used for high-volume production

where demand is so high that the product is continuously be-

ing produced. With mass production [55, pp. 83-106], equip-

ment is especially designed for the production of one

product, thereby eliminating much of the labor and material

handling costs generally incurred in the two previous types

of shop discussed.

There are two types of mass production: quantity and flow

production. Quantity production is generally invoked with a

product which requires a limited number of operations mak-

ing a bolt for example. Flow production refers to the pro-

Introduction 5

duction of an item which requires a long sequence of

operations ~ such as manufacturing an automobile.

1. 1.2 CLASSIFICATIONS BY PLANT LAYOUT

Production volume has a significant impact on the choice

of plant layout. Production volume, however, is not the only

factor. The physical properties of the product and material

handling concerns also affect the plant layout choice.

1. 1.2. 1 Fixed Position Layout

A fixed position layout [&MT., pp. 132-134] is used when

a product is so large that it , is not feasible to move it

through the shop. The· product remains stationary, and all

equipment is brought to it. This layout is used, for example,

in the manufacture of large aircraft and ships.

1.1.2.2 Process Layout

In order to allow fox; greater flexibility in machining

requirements for products, equipment is grouped by function

in a process layout [55, pp. 27-48]. Each function (or

process) is a separate department. Job.s must be routed from

department to department, throughout the shop, to be com-

Introduction 6

pleted. This allows for great flexibility in machining re-

quirements for products.

1.1.2.3 Product Flow Layout

In the product flow layout [ 55, pp. 83-106], jobs are

moved along a production line, often on a conveyor belt, with

machines adjacent to the line. This configuration is used for

high-volume production and is highly inflexible: once the

shop is laid out in this fashion, only_ an extremely narrow

range of products can be produced. Nonetheless, this layout

will allow the shop to produce the few products processed

highly efficiently.

1.2 ALTERNATIVE PRODUCTION SYSTEMS

1.2.1 ORDER POINT SYSTEMS

Material requirements planning was perhaps the first

production-inventory system used, albeit informally [47]. But

as products became more sophisticated during the first half

of this century, companies were no longer able to perform all

the calculations this approach requires. To handle this

emerging problem, order point systems were used. In an order

point system, the components necessary for production are

kept- in inventory. When the inventory is reduced to a speci-

Introduction 7

fied level, a pre-determined quantity of the item is ordered.

A reorder point is set for each item by determining the lead

time demand plus a safety stock to handle variability in lead

time and demand. An order quantity is often set using the

Economic Order Quantity (EOQ) or some sort of probabilistic

approach (14] [40]. Another approach to setting order guan-

tities is the ABC classification method in which the manager ,,

is urged to concentrate on reducing inventory for the rela-

ti vely small number of components which compose the bulk of

the company's inventory costs [67].

The most fundamentai problem with the order point approach

is that it assumes that there exists an independent re-

lationship (i.e. , there is no relationship) between the de-

mand for the various components heeded for production. There

is actually a dependent relationship between the demand for

components used in the production of a final product since,

for example, every time a car is ordered four tires, two

axles, one steering wheel and so for th are ordered. If the

demand for components were independent there would be no re-

lationship between these items. One implication of this as-

sumed independence is that large amounts of safety stock must

be carried for each item since if only one of the items needed

to produce the final product is out of stock, production must

halt. As the number of components used to make a product in-

creases, the probability of any component being out of stock

must decrease to keep the overall probability of stopping

Introduction 8

production constant, thus raising the level of safety stock

required for each component [62].

Another problem with order point systems is that they are

inappropriate for "lumpy" demand which causes large fluctu-

ations in daily demand. EOQ and many other techniques used

in lotsizing assume there is a continuous rate of demand for

the item - which is generally not a valid assumption. Thus,

the lotsizes determined by these methods are based on incor-

rect model of inventory costs. Finally, order point systems

do not explicitly consider capacity requirements or the tim-

ing of requirements necessary for demand to be met just as

needed [ 42].

1.2.2 MATERIAL REQUIREMENTS PLANNING

The objective of Material Requirements Planning (MRP) [42]

is to determine the requirements for meeting demand and to

generate the information needed for correct inventory

actions. MRP works backwards from a schedule of demand for

final products to determine the requirements for components

in terms of the quantity required and the date by which each

process must be completed.

There are three categories of inputs to an MRP system

[ 42]: a master production schedule, a bill-of-material file

and an inventory records file. The master production schedule

is a list of quantities and need dates for the final pro-

Introduction 9

---- ---~ ___ ,_ ---

ducts. The bill-of-material file is a list of the components

needed to make those products and data about the production

process necessary for each component. The inventory records

file contains the projected inventory balance for each com-

ponent and final product. In an MRP system, it is imperative

that the projected inventory balance be extremely accurate.

Many go as far as suggesting that all storerooms be kept un-

der lock and key to insure the accuracy of the inventory in-

formation [ 47].

Once the master production schedule is set, the gross re-

quirements for components are generated using information in

the bill-of-material file. A net requirements list is then

generated by subtracting the projected inventory balance

found in the inventory records file from the gross require ....

ments. Then working backwards from the final product, the

date when a process must be completed so the following proc-

ess will not be impeded is determined. The output for an MRP

system is a schedule indicating when each production process

must begin and the lotsizes that are required.

MRP systems have three major advantages over order point

systems [ 65]. First, using an order point system, safety

stock must be kept for each component to handle forec-ast er-

rors. This springs directly from the fact that an order point

system acts as if the demand for components of a final prod-

uct are independent. In MRP, safety stock for forecast errors

need only be carried for the final product - thus, in-process

Introduction 10

inventory for components

second advantage to MRP

can be significantly reduced. A

is that it is designed to handle

lumpy demand. Order point systems operate best when demand

is smooth [62]. Finally, MRP explicitly considers time phas-

ing of production. Order point systems are oblivious to time

phasing concerns.

MRP, however, is not without problems. MRP has been crit-

icized for contributing to long production cycles times be-

cause it builds exce~s safety lead times into the production

schedule. There is also a tendency to use large buffer in-

ventories in case there are problems such as machines break-

ing down or there are many defects. All this leads to

excessive in-process inventories [56]. A second fault of MRP

is that it does not deal with capacity planning; it is simply

assumed that the plant has the capacity to meet the schedule

established by MRP. Closed-loop MRP, sometimes called manu-

facturing resource planning (MRP-II), is an attempt to cor-

rect this fault by starting with a basic MRP system and

including capacity planning in a closed-loop or iterative

fashion [49] [65]. In a recent study by Anderson, Schroeder,

Tupy and White [ 1], it was found that only 9. 3 percent of the

firms that have implemented an MRP system are considered

class A users where scheduling and planning are performed in

a closed-loop system, inventory is under control and little

expediting is required. Sixty two percent of the users sur-

veyed are considered class C or D users where capacity plan-

Introduction 11

ning is done informally and expediting is used to control the

flow of work. Other research [38] indicates that of the com-

panies surveyed, only 44 percent of the companies using MRP

have found it to be cost effective and [58] only 50 percent

of the companies were satisfied with MRP.

1.3 JUST-IN-TIME

The concept of just-in-time (JIT) began over 20 years ago

in the Japanese shipbuilding industry [57]. Steelmakers had

overexpanded making it possible for shipbuilders to get fast

deliveries on steel orders. Taking advantage of this,

shipbuilders dropped their inventories from 1 month's supply

to a supply of 3 days. JIT ideas spread to other Japanese

companies who began demanding similar service from their

suppliers. Later, these companies began applying the same

concepts in their internal operations [57].

The objective of JIT systems is " ... to produce the neces-

sary units in the necessary quantities at the necessary time

[36]." JIT in a broad sense refers to" ... all the activities

of manufacturing which make the just-in-time movement of ma-

terial possible" [15] and not just the movement or transport

of material activities associated with the system. JIT sys-

tems are characterized by a fanatic obsession to reduce in-

process inventories. The Japanese consider in-process

inventory to be a waste of resources - but even worse, they

Introduction 12

claim that "inventory is the root of all evil [16]." It has

been shown that by reducing in-process inventory (by reducing

lotsizes) significant improvements in quality, worker moti-

vation and productivity can be obtained [57]. ,/

JIT received widespread attention during the oil shock of V

1973; when most Japanese companies lost money, Toyota showed

a huge profit using their just-in-time with Kanban system

[ 3 6] . In 1980, Toyota turned over their inventory every 4

days and reduced their break-even point to 64 percent of

sales. It was determined that Japan's cost advantage for a

comparable car was $1,700 during that time. The cost differ-

ence over United States' firms was attributed mainly to "ad-

versarial labor relations, excessive inventories, lagging

productivity, and inferior quality performance [7]."

During the 1970's, many Japanese manufacturers switched

to using JIT systems. Now, during the 1980's, many American

firms are embracing JIT techniques. The list of American

companies using JIT includes: Chrysler, American Motors,

Hewlett Packard, Apple, Burroughs, Black and Decker, Bendix,

GM, Ford, Kawasaki and Harley Davidson [6] [57] [59]. Using

JIT concepts, Harley Davidson reduced their break-even point

by 32 percent, reduced defects by 24 percent, reduced in-

process inventory from $23 million to just over $8.5 million

and significantly reduced stockouts I 59] . Such performances

have led the American Production and Inventory Control Soci-

ety to call for a Zero Inventory Crusade. (Note that the

Introduction 13

terms Zero Inventory and JIT may be used interchangeably

[ 66] ) .

1.3. 1 CULTURAL INFLUENCES IN JAPAN

1.3. 1.1 Respect for Humanity

The Japanese use a management by consensus approach [22]

[32] [43] [45] [53]. They spend a great deal of time getting

everyone involved in the decision making process. Although

much time is spent obtaining a consensus, once it is reached

the plan is implemented more rapidly since everyone is com-

mitted to the plan. The Japanese also place a strong emphasis

on keeping the lines of communications open within the com-

pany. An example of this which has received considerable

attention recently is the guali ty circle [ 8] [ 9] [ 50]. A

quality circle is a small group of workers in a shop or de-

partment that meet voluntarily to discuss how to improve

product guali ty. Some of these groups take a broader per-

spective and include other considerations such as productiv-

ity improvement; these groups are sometimes called

participatory circles. The scope of these groups is limited

to improvements within their own group or department.

Another aspect of respect for humanity is" ... allying hu-

man energy with meaningful, effective operations by abolish-

ing wasteful operations [36 p.125] . 11 It is felt if a worker

Introduction 14

sees the job as important, motivation and self-esteem will

be high. If the worker feels the job is wasteful or insig-

nificant~ morale and self-esteem will suffer.

1.3.1.2 Lifetime Employment

Major Japanese companies have a tradition of lifetime em-

ployment for their workers. Only in extreme circumstances

··will these companies lay off workers. This has many impli-

cations for the production system. First, the workers tend

to identify more with the company and see a link between

their success and the company's success - improving worker

morale. Also, since the turnover rate is low, the benefits

of employee training programs accrue over a longer period of

time.

1.3.1.3 Vendor Relationships

Since Japan is a small country geographically, vendors are

often located near the companies they serve. This is impor-

tant because in JIT systems vendors are generally expected

to make three or four deliveries a day on demand. Most United

States companies receive weekly or monthly deliveries from

their suppliers. Many United States companies implementing

JIT systems have asked their vendors to relocate closer to

their factories.

Introduction 15

Another difference in vendor relationships is the way

American and Japanese companies manage their vendors [ 12]

[ 56] . American companies tend to view vendors as adver-

sari es. They generally use multiple sources for i terns and

tend to play the vendors against each other. The Japanese

deal with vendors in a totally different manner. Vendors are

considered as co-workers and are treated as an extension of

the factory. The companies tend to have long term relation-

ships with their vendors.

1.3.2 ASSUMPTIONS NECESSARY FOR JIT SYSTEMS

1.3.2.1 Small Setup Times

The major thrust behind JIT systems is to reduce inventory

by producing small lotsizes. To do this economically, it is

necessary to reduce setup times. The Japanese try to reduce

setup times to under ten minutes. On achieving this they try

to reduce setup times to under one minute; this is called a

"one touch setup''. Frequently it takes American companies

from hours to entire days to P.erform setups on similar ma-

chines [37].

The Japanese attack the setup problem in several ways

[ 3 6] . They do not use specialists to perform setups. Line

workers are expected to operate and setup a variety of ma-

chines. When a setup is required, the workers get together

Introduction 16

-----~ -

to setup the machine in a parallel fashion, removing the

bottleneck the setup creates. Another way they reduce setup

times is by designing their own machines using in-house en-

gineering staffs. The machines are designed specifically to

reduce the time needed for setups. A third way they reduce

setup times is by separating the actions required in setting

up a machine into two categories: the external setup and the

internal setup. The external setup is the part of the setup

which can be done while the machine is running. The internal

setup is the part of the setup which can only be done when

the machine is stopped. For example, getting the tools re-

quired to perform a setup is part of the external setup since

it can be done while the machine is running. Conversely,

changing a drill bit can only be done only after the drill

press is stopped - so it is part of the internal setup. What

the Japanese try to do is to make as much of the setup as

possible part of the external setup and make sure all the

external setup is performed before the machine finishes a

job. This reduces machine downtime during a setup since none

of the external setup is performed while the machine is idle.

1.3.2.2 Frozen Demand Schedule

JIT systems, according to the literature, require that

production schedules be essentially frozen for about a month

[15] [36]. Once the production schedule is set, the lines

Introduction 17

/ \

are balanced. Deviations from this production schedule may

cause the lines to get out of balance - causing the pro-

duction to back up. Toyota has found that its JIT system can

handle demand fluctuations of up to 10 percent by adjusting

the length of the workday. The workers stay unti 1 the work

is done. JIT systems purportedly cannot handle larger f luc-

tuations in demand. Many companies smooth demand by filling

orders from the previous week. The actual daily production

schedule using this approach is the average daily demand for

each product from the previous week. Another approach is to

use . small buffer inventories of popular product lines to

smooth production. Companies draw from this inventory or

build it up (by producing less or more than the actual demand

for that day respectively) to smooth' the production schedule.

This practice, however, should be kept to a minimum since it

defeats the purpose of Just-in-Time production.

1.3.2.3 Workforce Attributes

The Japanese place a great emphasis on training their em-

ployees. This is vital in JIT systems because workers must

be cross-utilized extensively ~nd variability in processing

times must be kept to a minimum [21]. As mentioned earlier,

workers must be able to perform setups on a wide variety of

machines. JIT systems can be thought of as conveyor lines

running throughout the factory. When one workcenter takes

Introduction 18

longer than its allotted time processing an item, the entire

line must stop until it finishes.

1.3.2.4 Quality Control

Quality control is essential to a JIT system [ 24] [ 57].

Since there is no buffer inventory, when defects are found

the workers must go back to the preceding stage to have the

defect-producing worker fix it - stopping the entire line.

The Japanese do not use statistical quality control with

their JIT systems; they use what they call total quality

control which aizns/£0~ ze;i:-6d~fects. At many of these facto-/"" '--<'" ___ // . ries /d~fec1ts-"~re measur~d ~n parts per million. At United

- . , _ ____.,..rl . \\__; __ ,..,/-' ·-"-:=--=.-:-_-,.. -_-:--~~---:_---~- --·~ -States factories defects are ig_eJ1_§_:i;:_alJ_y measured in. parts per

hundred. /·'' -~-~- ·''"" -.,

.Quali ty\c:=ontrol is the responsioi\l:~:SY _of the worker on the .__ --\.. \ -~'~..___"---,._/~--,-:::-:-.~:~-

p todu_cti on line in JIT systems; companies with JIT systems \ . . .- --" .. :::::::::--, generally have small'\ qu1~l1~y cont;ol staffs. They also gen-

.! ':

erally do not have rewo;rk lines which in many United States

factories take up from 15 to 40 percent of total machine ca-

pacity in the plant [57]. It is the responsibility of the i --~' '

worker making the defect to correct hi~ or her mistake. This

is good from a behavioral view .because the worker and the

rest of the factory get almost immediate feedback about de-

f ects [ 56] . In an MRP system:, defects are thrown into a re-

work bin and a replacement is drawn from inventory. This does

Introduction 19

not provide the worker with the kind of feedback that a JIT

system supplies.

Another aspect of the quality control effort is preventive

maintenance, which the Japanese practice religiously. Not

only does this help reduce the number of defects, but it re-

duces the amount of machine downtime. In a JIT system, ma-

chine downtime must be kept to a minimum since there is no

i.nventory available to keep the line.going while the machine

is being repaired.

1.4 TOYOTA'S KANSAN SYSTEM

The Kanban system is a subsystem of Toyota's JIT pro-.

duction system. Monden [36, p. 41 describes the role of the \.

Kanban system in Toyota's production system in the following

manner:

Many people call the Toyota production system a Kanban system: this is incorrect. The Toyota production system is the way to make products, whereas the kanban system is the way to manage the just-in-time production method. In. short, the Kanban system is an· information system to harmoniously control the production quanti-ties in every process.

The Kanl::>an system is just a system of cards to control

production and inventory. There are primarily two types of

Kanbans: production Kanbans and withdrawal Kanbans. Another

key element in Kanban is the contain'er. A container is the

basic unit of production, ro'1:ghly analogous to a lotsize in

Introduction 20

MRP. The size of a container (the number in the lot) is kept

constant within components used for a particular product.

Kanban can be understood by examining a single workcenter

(stage N) within a factory [21) [35). All the containers at

stage N that are currently being processed or are in finished

inventory must have a production Kanban attached. Similarly, \

all containers in raw material (or pre-process) inventory

must have a withdrawal Kanban attached. For the succeeding

stage to get its pre-process materials from stage N, a with-

drawal Kanban must be presented at stage N, where a container

is pulled from finished inventory. The production Kanban is

removed from the container and is replaced with the succeed-

ing stage's withdrawal Kanban; the container and withdrawal

Kanban go to the succeeding stage. Thus, the withdrawal

Kanban authorizes the withdrawal of a container by the sue-

ceeding stage. The production Kanban is sent back to the be-

ginning of stage N - authorizing the removal of a container

of stage N's pre-process inventory with a withdrawal Kanban

·attached. The withdrawal Kanban is removed from the con-

tainer and replaced by the production Kanban which just ar-

rived at the beginning of stage N. This production Kanban

authorizes processing of the container. The withdrawal Kanban

is sent to the preceding stage which initiates replenishing

the pre.,.. process inventory at stage N.

In MRP, each workcenter is given a schedule of what to

produce, in what quantity and when it is needed. In Kanban,

Introduction 21

this information is transmitted by giving the final

workcent,er the Kanbans needed for meeting that day's demand.

The other workcenters get their production instructions by

.receiving withdrawal Kanbans throughout the day. This is why

Kanban is considered a "pull" system - products are pulled

through the factory 11 just in time"·to meet demand.

To summarize how Kanban works at Toyota; there are four

basic rules {15]: 1) every container holding parts must have

a Kanban attached; 2) containers are never transported be-

tween workcenters unless a wi thd·rawal Kanban is attached; 3)

production is always done in standard ·container sizes and;

4) production must not be performed unless authorized by a

production Kanban. Note that a production Kanban will always

stay within a workcenter. A withdrawal Kanban travels between

two workcenters.

There are other types of Kanbans, such as signal Kanbans

and vendor Kanbans [ 36] . Signal Kanbans are used when setup

times cannot be reduced to acceptable levels forcing Toyota

to produce lotsizes of several containers at a time. Vendor

Kanbans are essentially withdrawal Kanbans that travel be-

tween the vendor and the stage which uses the vendor's prod-

uct as their raw material. This illustrates a point

previously mentioned: vendors are treated as co-workers and

are considered an extension of the factory.

What Kanban attempts to do is to apply the principles of

a flow shop to the production of small batches. This is a

Introduction 22

I I •

compromise between the process layout and the product flow

layout called the group technology layout [ 13] [ 51]. This

layout places a series of machines together which perform

different functions - creating something similar to an as-

sembly line for a particular family of products. This allows

a series of processes to be totally automated while retaining

much of the flexibility of the process layout. The Japanese

have made large gains in productivity and have significantly

reduced lead times using group technology.

JIT systems are essentially order point systems. So why

does Kanban work when order point systems do not (12]? With

its sequencing of products and small lotsizes, Toyota is able

to achieve a fairly continuous demand for all the component

parts that order point systems assume. This also creates a

steady flow through all the workcenters which simplifies ca-

pacity planning to the extent that it can be performed by the

foreman at each workcenter. Also, since lead times are

short, the scheduling of jobs (or time phasing of require-

ments as it is called in MRP) is not crucial.

1.5 COMPARISON OF KANSAN WITH MRP

A primary objective of Kanban and MRP is to help firms

better manage their in-process inventory. The way the two

systems view in-process inventory is totally different. The

Japanese view in-process inventory as a waste of resources.

Introduction 23

l

They try to totally eliminate it. In MRP, large in-process

inventory is used to safeguard the factory from machine

breakdowns, late shipments, defects and so forth. This is why

MRP. is. sometimes called a "just-in-:-case" system [ 41]. The

tools the two systems use could hardly differ more [12]. MRP

uses a computer to crunch through a long sequence of compu-

tations - generating reams of output. Kanban controls inven-

tory by having workers exchange cards. In MRP, mathematical

and statistical techniques are generally used to calculate

lotsizes. In Kanban, inventory levels are set using a trial

and error approach. Kanbans are taken away from workcenters

(reducing inventory) until the shop starts to back up.

Using Kanban leads to significant reductions in inventory

1.nvestment, leadtime and stockouts - which MRP promises but

frequently does not.deliver [15] [36] [57] [59]. JIT systems

have a proven advantage in quality control [56]. Kanban af-

fords the user a better method of keeping track of in-process

inventory. Every i tern of inventory has a Kanban attached.

One of the major problems with MRP is keeping an accurate

accounting of in-process inventory [ 11] [ 27]. Another ad-

vantage of Kanbans is that losses from obsolescence are gen-

erally less than in an MRP system [36]. This is because there

will be much less inventory on-hand in a Kanban system when

model changes are made.

A more subtle advantage of Kanban is that it exposes

problems in the manufacturing process which can lead to im-

Introduction 24

provements. For example, suppose a machine has a large setup

time. The workcenter will have to use Signal Kanbans to

produce lotsizes of several containers. The increased inven-

tory at the workcenter will be extremely noticeable, signal-

ing there is a problem with this machine. This will put

pressure on everyone to resolve this problem. In MRP the an-

swer would probably be to increase the lead time at that ma-

chine and perhaps increase the lotsize. Thus, the tendency

in MRP is to work around problems rather than to solve them.

MRP is believed to apply to a wider range of manufacturing

environments. The assumptions for using Kanban are much more

restrictive. Achieving a near frozen production schedule is

hard to do in practice. For example, product options for cars

can cause problems when attempting to smooth the production

schedule. A particular make of car can have many combinations

of options - each combination is a different product. To

handle this, Toyota offers a limited number of option pack-

ages. This not only reduces the number of different products,

but it stabilizes daily demand. Of course, this does have

marketing drawbacks since Toyota can not cater as much to the

individual customer. The assumptions with regard to imple-

menting Kanban (reducing setup times, reducing the variance

of processing times and cross-utilization of workers) are

achievable, but may require substantial amounts of time and

money. However, one only needs to look at almost any issue

of ;Production and Inventory Management ([2] [4] [28] for ex-

Introduction 25

ample) to see that implementing MRP is probably at least as

difficult and expensive as implementing Kanban.

1.6 REVIEW OF QUANTITATIVE RESEARCH IN KANBAN

Osaka and Terada I 25] compared the amplification of pro-

duction and inventory fluctuations in a pull system (such as

Kanban) and a push system using analytical models of the two

systems. They found that when container sizes in a pull sys-

tem were small, amplification of production fluctuation did

not occur in the succeeding workcenter. However, using

larger container sizes, amplification of production fluctu-

ation in the succeeding workcenter was quite large. The other

factor found to cause amplification in a pull system was

large lead times for jobs at a workcenter. Amplification of

production and inventory fluctuations in push systems was

found to be caused only by errors in forecasting demand.

Huang, Rees and Taylor [20] [21] examined how adaptable a

JIT with Kanban system is to a United States production en-

vironment via a Q-GERT simulation model of a multi-stage flow

shop. Their findings show that variability in processing

times and demand rates results in an increase in overtime in

their sample shop. It was found that by increasing the number

of Kanbans, these effects could be reduced. However, it was

shown that an increase in Kanbans would not reduce the ef-

fects of bottlenecks in the production line. Their conclu-

Introduction 26

sions are that a company considering making the change to a

Kanban system should be prepared for a lengthy transition

period. They estimate that it would take at least one year

to train workers and standardize machine processing times and

setup times.

Krajewski, King, Ritzmann and Wong (26] compared Kanban

to MRP using their Manufacturing Simulation System (MASS)

package. They found that Kanban does well (better than MRP)

in the more favorable manufacturing environments they

studied. However, they found that an order point system

worked equally well in the same environments. MRP, however,

appeared to be the more robust system. Their conclusions were

that Kanban is a good way to implement small lot production

and to expose environmental problems which, if fixed, would

lead to more efficient production.

Rees, Huang and Taylor [ 52] also compared MRP to Kanban

for a shop with both serial and assembly operations using two

Q-GERT simulation models. (However, they only examined JIT

without group technology applied.) They found in their sam-

ple shop that MRP could be converted to a Kanban operation

by reducing leadtimes and setup times, but that more savings

would be obtained if the shop were left as an MRP shop and

the same leadtime and setup reductions were effected.

Introduction 27

1. 7 MOTIVATION AND SCOPE OF INVESTIGATION

The current quantitative literature on JIT can be divided

into two categories. The first category examines the opera-

tion of a JIT system in a flow-shop environment which does

not possess the unique ·aspects generally found in Japanese

production shops. The second category compares JIT with MRP

within specific sample shops. The thrust of the current

literature is to transfer a JIT system as is, into a diffei-

ent environment and see how well it works. The investigation\\

in this disser-tation differs from those in the current lit-\\ \\ \I erature in that the focus here is on how to adapt a JIT system ;

so that it works well in suboptimal environments rather than (

simply to transfer a JIT system into a different environment~.

and see if it works. For example, the literature indicates

that machine setup times must be drastically reduced in order

to use the standard JIT with Kanbans system. Part of this

investigation focuses on how to adapt a JIT system so that

it will work even if setup times cannot be reduced enough to

allow for standard JIT with Kanban operations.

As stated above, the purpose of this dissertation is to 1

I explore adapting JIT so that it will work when less than \ ~

ideal settings and conditions are in effect. These suboptimal

conditions are very likely to exist in American production

environments that are considering implementation of the JIT

system with Kanbans. Specifically, this dissertation exam-

Introduction 28

i~es three important problems faced by an American firm im-

plementing a JIT system with Kanban: what should be done if

setup times cannot be reduced to a level that allows small

lot production (one container lots) at all workcenters; what

factors in an American production environment influence the

number of Kanbans required at a workcenter, and how should

the number of Kanbans at a workcenter be determined; and, how

should the number of Kanbans at a workcenter be dynamically

adJ"usted when conditions in the shop are not stable. u /

In particular, Chapter 3 addresses the problem of using,/

JIT with Kanbans when the firm cannot reduce all machine

setups to acceptable levels. The use of Signal Kanbans, which

facilitates lot production within a Kanban framework to han-

dle this problem, is explored. The investigation examines

setting lotsizes for the Signal Kanbans and the feasibility

of using Signal Kanbans versus standard Kanbans in an Ameri-

can production environment. I

Chapter 4 investigates how various factors that are pres~v

ent in the typical American production environment influence

the number of Kanbans that are necessary for smooth operation

of the shop. First, a descriptive model of the relationship

between the various shop factors and the number of Kanbans

is developed and tested. Second, a methodology for determin-

ing the initial number of Kanbans is presented and tested.

Finally, the issue of the extent to which shop conditions can

deteriorate and JIT still work is addressed.

Introduction 29

Chapter 5 addresses whether JIT will work if the shop en-

vironment is not as stable as in Japanese shops. A methodol-

ogy for dynamically adjusting the number of Kanbans in

response to unstable shop conditions is presented. The meth-

odology is tested over a wide range of conditions in the

sample shop.

In summary, this dissertation investigates the implemen-

tation of JIT in a non-Japanese production environment and 11;./ how J'IT should be adapted so that it can have a broader range

of applicability. This research should be of interest to

American production managers considering the implementation

of JIT in their factories. Each of the three parts of this

investigation addresses important concerns facing a firm im-

plementing JIT in suboptimal environments.

Introduction 30

2.0 THE EXAMPLE SHOP

A simulation model of an example shop is used throughout

this investigation to verify the analytical results developed

in each of the next three chapters. The example shop, the

product structure and the Q-GERT model used throughout the

dissertation are described in this chapter.

2.1 THE PRODUCT STRUCTURE AND EXAMPLE SHOP

The product structure and example shop used throughout·

this investigation are illustrated in Figure 2-1 and Figure

2-2, respectively. The shop has the characteristics of both

a job shop and a flow shop. This configuration was chosen

because most American shops usually combine assembly and se-

rial operations. Almost all quantitative JIT research to

date has focused on pure flow shops. The choice of shop

configuration furthers one of the major aims of this re-

search: to investigate JIT in an American production envi-

ronment.

Workcenters l, 2 and 4 have the characteristics of a job

shop. A variety of operations are performed at these

workcenters. Workcenters 3, 5 and 6 are assembly workcenters.

Workcenter 3 combines two B's and three H's to make an E,

three A's to make a J, two C's and three K's to make an I and,

The Example Shop 31

x I -

E J .

tb3 F 2 A

------8 2 3

y

I

I I M

c 2 K 3 L 2 D 2

Figure 2-1. The Product Structure.

32

w w

I

2

ABCD' E

F,L

Figure 2-2. The Example Shop and Item Flows.

5

6 ...--.. y

two D's to make an M. Workcenters 5 and 6 are different from

the other workcenters in several respects. They only produce

one product each, so they never incur setups. Also, these two

workcenters have only one machine whereas the other four

workcenters each have two machines. Workcenter 5 assembles

one E, two F's and one J to make an X. Workcenter 6 assembles

one I , two L' s and one M to make a Y. Further detai 1 s of the

shop such as demand for products and costs of production vary

from chapter to chapter and hence are provided when appro-

priate.

2.2 THE SfMULATlON MODEL

The example shop is investigated using a Q-GERT model.

Q-GERT [48) is a network oriented simulation language written

in ANSI FORTRAN. Sixty user written routines (approximately

2000 lines of FORTRAN) are used in addition to the network

model. The model keeps track of inventory, setup and backor-

der statistics. Moreover, the total cost of production is

computed and di splayed along with its components: inventory

costs, backorder costs, and sh9rtage costs. The basics of

the model can be explained by describing one assembly

workcenter (workcenter 6) and one fabrication workcenter

(workcenter 4) since the rest of the workcenters are simi-

larly constructed.

The Example Shop 34

.i

2.2.1 THE ASSEMBLY WORKCENTER

The simulation model for assembly workcenter 6 is illus-

trated in Figure 2-3. Orders are generated by user function

70, which is called whenever Node 180 is realized. The user

function places a transaction representing an order (or a

withdrawal Kanban) into Queue Node 80. Finished goods inven-

tory is stored in Queue Node 86. Whenever Queue Nodes 80 and

86 contain transactions at the same time, Assembly Node 88

is realized. This is analogous to an order and a final prod-

uct being matched. Upon matching, a transaction is sent to

Node 89 and then to Queue Node 2, the production ordering

post. This signifies that the workcenter should start pro-

duction for Y as soon as the machine is available.

Whenever Nodes 64, 74 and 84 are realized, the full re-

quirement of I, L and M, respectively, to make a product Y

has arrived. It takes two transactions to realize Node 74

since it takes two L's to make a Y. Only one transaction is

needed to realize Nodes 64 and 84 since product Y only re-

quires one I and one M. Upon being realized, Nodes 64, 74

and 84 send transactions to Queue Nodes 81, 82 and 83, re-

spectively. When Queue Nodes 81, 82 and 83 all contain

transactions (which means that all the pre-process materials

needed to make a Y are present) and Queue Node 2 contains a

transaction (signalling that an order has been placed for a

Y) the production order and the pre-process materials are

The Example Shop 35

U•IU931fJOM lu!pa.fd

WOJ;I •PJOeH

ue

0 uoGuo)t --~· . (JJ llOJ fDMOJpllfM

\

matched by Assembly Node 87 which causes a transaction to be

placed in Queue Node 85 in front of the machine.

When the machine becomes available, User Function 8 is

called. This causes a transaction to be placed in Nodes 96,

97, and 98 using Subroutine PTIN. From here the transactions

go to the appropriate preceding workcenters. This represents

the workcenter ordering additional containers of pre-process

materials that are depleted by starting production. Then, a

service time is assigned for the production activity which

spans Nodes 85 and 186. When Node 186 is realized, a trans-

action is sent to Queue Node 86 signitying that a new Y has

been produced and put into inventory.

2.2.2 THE JOB WORKCENTER

Workcenter 4 performs job-shop type operations on compo-

nents F, J, L or M. The simulation model for job workcenter

4 is illustrated in Figure 2-4. Orders for finished goods

at workcenter 4 come from workcenters 5 and 6. Each of these

is represented by a transaction being placed in Queue Node

40. Transactions representing the workcenter' s finished

goods inventory are stored in Queue Node 46. Attribute l

identifies the type of job the transaction represents, i. e,

whether the component to be processed is an F, J, L or M.

When transactions that have the same attribute 1 value are

in Queue Nodes 40 and 46 at the same time, they are matched

The Example Shop 37

w co

Mokh on All I

lJf 21 Uf4

Figure 2-4. Job Workconlor 4

Wlthdrowol l<onbon Poal

Molch on AHi

Reorder from PrecedlnQ Wortant ...

/fl 49

by Match Node 48. A transaction representing the filled order

is sent to the ordering workcenter. Another transaction is

sent to Queue Node 41, the production ordering post. The

transactions in Queue Node 42 represent the pre-process ma-

terials for workcenter 4. When transactions that have the

same attribute l value are in Queue Nodes 41 and 42 at the

same time, a transaction is placed in Node 145. When Node 145

is realized, User Function 21 is called. Here it is deter-

mined which of the workcenter's two servers will get the job

finished quicker using a minimum throughput rule. If the

first server is chosen, the transaction is sent to Queue Node

45 by placing a 1 in attribute 2. Likewise, a 2 is placed in J

attribute 2 if the transaction is to be sent to Queue Node

147. Node 145 branches on attribute 2, so the transaction is

sent to the proper queue. When the server becomes available,

User Function 4 is.called. This causes the pre-processed ma-

terial that is being used to be replaced by putting a trans-

action in Node 47. From there, an order is sent to the

appropriate preceding workcenter. Then, User Function 4 as-

signs the service time from the appropriate distribution.

When the activity is completed, Node 146 or 148 (depending

on which server handles the transaction) is realized. Then,

a transaction is sent to Queue Node 46. This represents the

finished goods inventory at workcenter 4.

The Example Shop 39

2.3 SUMMARY

The example shop, the product structure and the Q-GERT

simulation model used throughout the next three chapters have

been presented.

and setup times,

Shop parameters, such as processing times

vary throughout the investigation and are

described in the following chapters for each set of simu-

lation runs.

The Example Shop 40

3.0 DETERMINING WORKCENTER LOTSIZES FOR SIGNAL KANBANS

Japanese experiences have implied that small setup times

relative to processing times are essential in a JIT pro-

duction system with Kanbans. Without such small setup times

large bottlenecks at workcenters resu1 t and the production

operation becomes hopelessly delayed. The inability to sig-

nificantly reduce setup times prohibits many American firms

from attempting to use the JIT technique with I<.anbans espe-

cially if the production system is a non-repetitive job

shop-type operation rather than an assembly line-type opera-

tion. American firms are often unable to make the necessar-

ily large investment in new machinery or extensive worker

training required to reduce setup times. However, the Toyota

Company [ 36] [ 3 7] has developed a novel means for employing

the JIT technique with Kanbans for an operation that encom-

passes several workcenters that have large setup times rela-

tive to processing times (such as a forging process or punch

process) . In such a system a special type of Kanban, re-

f erred to as a "signal Kanban" is used in conjunction with

the workcenter or process that has relatively large setup

times. In effect, a signal Kanban triggers the production

of larger than normal lots at workcenters with large setup

times within a JIT framework, wherein standard Kanbans at

normal workcenters concurrently trigger the production of

Determining Workcenter Lotsizes For Signal Kanbans 41

containers encompassing only a very small number of uni ts,

hopefully only one.

This altered approach developed by the Japanese for em-

ploying the JIT system using both standard and signal Kanbans

offers an alternative for American firms who desire to im-

plement a JIT system with Kanbans, but, are unable to reduce

setup times at selected workcenters to a feasible level, at

least initially. However, this altered version of the tra-

ditional JIT system requires that appropriate lotsizes be

determined to be used in conjunction with the signal Kanbans.

Failure to achieve effective lotsizes can result in large

backorders and the inability to meet demand. For reasons

that will be presented later in this chapter, the traditional

multi-product EOQ lotsizing approach does not always work for

signal Kanbans. As such, the purpose of this chapter is to

demonstrate the use of integer mathematical programming for

determining the optimal lotsizes to be used in conjunction

with signal Kanbans in a JIT system. Specifically two inte-

ger programming model versions are developed and solved, one

of which minimizes inventory at a workcenter while the other

model version minimizes inventory and setup costs. Both

models encompass constraints that prohibit backorders ·and

require demand to be met. Preceding the development of these

two integer programming models, the multi-product EOQ

lotsizing approach is tested using a Q-GERT simulation model

of an example shop. This same simulation model and example

Determining Workcenter Lotsizes For Signal Kanbans 42

is subsequently used to test the two integer programming

lotsize models, and, to explore some of the characteristics

that an American firm might encounter in the implementation

of a JIT system with both signal and standard Kanbans.

3.1 THE JUST-IN-TIME SYSTEM WITH SIGNAL KANBANS

The Japanese control the stage-to-stage authorization of

container production with two "Kanbans," which are simply

cards. One card, called a production Kanban, accompanies the

containers as they are being produced, as shown in Figure

3-la. Looking specifically at stage N in Figure 3-la, when

the production of a container is completed and demand from

the succeeding stage (N-1) occurs (as indicated by a with-

drawal Kanban from stage N-1), the production Kanban is re-

moved from the container and is returned to the

production-ordering Kanban post at the same stage (N). The

withdrawal Kanban from stage N-1 actually replaces the pro-

duction Kanban on the container, and it accompanies the con-

tainer to stage N-1. For production activity to take place

at stage N, both a production Kanban and a container of the

required parts accompanied by a withdrawal Kanban must be

present 'at that stage. The production Kanban subsequently

replaces the withdrawal Kanban, and the withdrawal Kanban is

sent back to stage N+l where it authorizes stage N+l (the

preceding stage) to produce another container now required

Determining Workcenter Lotsizes For Signal Kanbans 43

I I I I I I I I I I

D -----w---

~' 0 WITMOlt.&ftl, 11.&IWeM

0 ll"<!OUCTIO• 11.&•e&11

• ....TllC.&1. UlllT

Figure 3-la. The Nol:mal Kanban Ope~a~ion.

--------------------------------------~

COllT.&INElt 1 COllTAlllElt I ~OUCTION ACTIVITY CCNTHCll 2. COlllT'Al•Elt 2 ---, - ] - ]

: ~ .... ' ..., '

C*TAINElt 5 C*TAINElt .._.. .. , ... -.. , - C*TAINElt

,------v I • 1..., ___ _

--...1 CDNTAINElt 1 ,_ CZ*TAINElt

1...---------............

--------------------~~

Figure 3-lb. The Kanban Operation with a Signal Kanban.

44

at stage N. This creates a continuous cycle of container

movement between the stages.

In effect, the Kanbans "pull" the containers through the

production system "just-in-time" to meet demand at each pro-

duction stage, thus minimizing in-process inventories. In

this process two Kanban swaps are made; one immediately prior

to the production activity and one immediately following the

production activity at each stage. The production Kanban

never leaves its "home" stage while the withdrawal Kanban

moves between stages.

intraprocess control

The production Kanban acts

apparatus and the withdrawal

serves as the interprocess control apparatus.

as an

Kanban

The Japanese success with the JIT system with Kanbans is

attributable to several factors, most prominent of which are

the small setup time, highly skilled and trained Japanese

worker, and job automation. These factors enable Japanese

companies to achieve small and almost constant processing

times with very little variability. Such scheduling systems

are further enhanced by extremely close cooperation between

the supplier, manufacturer and customer that minimizes vari-

ability in input rates and demand schedules.

When the production system consists almost exclusively of

workcenters at which small setup times (relative to process-

ing times) cannot be achieved, such as exists in many Ameri-

can job shops, large bottlenecks occur at these workcenters

in the JIT system, which in effect, causes the system to

Determining Workcenter Lotsizes For Signal Kanbans 45

collapse. However, in batch-mode production that encompasses

a number of workcenters with small setup times and a limited

number of workcenters with large setup times, the JIT system

can still be employed with the use of "signal" Kanbans.

The JIT system with a signal Kanban is illustrated in

Figure 3-lb, which is actually an overlay of the correspond-

ing section in Figure 3-la. (The signal Kanban system em-

ployed in this chapter is more efficient than several other

alternative signal Kanban systems available. The interested

reader can see Monden [36] for an alternative approach.) The

workcenter depicted in Figure 3-la produces two items and the

setup times for the two i terns are large relative to the

items' processing times. In this scenario, both stages N-1

and N+l operate with normal Kanbans while the workcenter at

stage N (in Figure 3-lb) employs a signal Kanban. For exam-

ple purposes this workcenter is shown with two "stores" for

i terns 1 and 2, respectively, each containing a lotsize of

seven containers.

Since· stage N-1 is a standard Kanban workcenter, when a

container of· i tern 1 is needed at stage N-1, a withdrawal

Kanban is sent to stage N and the top container (labelled

number 1) is taken. Now six containers exist at the store

for i tern ·1 in stage N. In the standard Kanban process when

a withdrawal is made at N, a production Kanban would be sent

back to the beginning of stage N to trigger the production

of another container of i tern l, (which would create setups

Determining Workcenter Lotsizes For Signal Kanbans 46

as production switches for each container from one i tern to

the other). However, when using a signal Kanban no pro-

duction Kanban is seht back to the beginning of stage. N --

yet.

As stage N-1 continues to send withdrawal Kanbans for item

1 to stage N, the supply of containers from the store of item

1 will continue to be depleted until the container with the

"material" Kanban denoted by the rectangular designation (Ci

is reached. When this material Kanban is reached a pro-

duction Kanban is still not released, but this particular

Kanban is a warning that the lot is running low, and, a

withdrawal Kanban is sent back from stage N to stage N+l for

a container of the required item produced at stage N+l. Al-

though, when this container reaches stage N there will be no

production Kanban to trigger production, there will be a

container ready for the imminent arrival of the signal Kanbah

which initiates production of a lot.

The next container withdrawn from the store at stage N

contains the signal Kanban designated by an inverted triangle

(V). As this container is withdrawn, the signal Kanban is

sent to the beginning of stage N to 11 signal 11 the ihi ti a ti on

of the production of a lot. Since required items were pre-

viously ordered from stage N+l, production can start imme..,.

diately. However, note that there is only one container of

in-process i terns at the beginning of stage N. The signal

Kanban also authorizes the production of multiple containers,

Determining Workcenter Lotsizes For Signal Kanbans 47

I (\J !

i I

i I r:

-----~-------------------------------'--------- - ----- - --------~ -- ~ - -

in this case a lotsize of seven containers, at stage N+ 1.

As soon as production begins at stage N, a withdrawal Kanban

is sent to stage N+l for the next container and the normal

Kanban process operates between stages N and N+l until the

lotsize of seven containers is completed.

As a result of this large lot, it is unnecessary to make

a production run for each withdrawal Kanban from N-1, thus

eliminating setups. Al though the use of such buff er inven-

tory lots is inconsistent with the Japanese philosophy of

inventory reduction embodied in the JIT system, it is an op-

erational compromise that enables a company that does not

totally meet the requirements for JIT implementation to reap

some of the benefits of the JIT system.

A crucial aspect in the implementation of signal Kanbans

is the determination of the lotsize used in conjunction with

the signal Kanban. A lotsize larger than necessary will

needlessly increase inventory costs, thus offsetting the ob-

jective of employing the JIT system, While a lotsize too

small will incur excessive setup costs and may create back-

orders that will ultimately cause a JIT system to completely

collapse. As such, the purpose of this chapter is to demon-

strate how an integer mathematical programming model can be

developed that will determine the optimal lotsize to be used

in conjunction with a signal Kanban in a JIT system. How-

ever, prior to proceeding with this model development, a c~se

scenario of an example shop using a JIT system will be de-

Determining Workcenter Lotsizes For Signal Kanbans 48

scribed in order to test the classical EOQ approach to

lotsizing, and, to subsequently test the mathematical pro-

gramming modeling approach.

3.2 CASE EXAMPLE AND SIMULATION MODEL OF A SHOP OPERATION

The case example that will be employed to analyze the

lotsizing models encompasses a production operation with six

workcenters that produce two products, X and Y. The product

structure for each product is shown in Figure 2-1. The con-

figuration of the six workcenters in the example shop is il-

lustrated in Figure 2-2. This shop has been given the

characteristics of both a job shop and an assembly shop in

order to explore the feasibility of the signal Kanban system

in the broadest context possible. At workcenters l, 2 and

4, which have the characteristics of a job shop, a variety

of operations is performed. However, workcenters 5 and 6

differ from the other workcenters in several respects. They

only produce one product apiece so they never incur setups,

and, these two workcenters have only one machine whereas the

other four workcenters each have two machines.

Of the four workcenters that perform operations requiring

setups only workcenter 1 has high setup times relative to

container processing times. Workcenter 1 requires 0.5 hours

to setup for any of the four i terns, A, B, C and D, produced

at that workcenter. Container processing time is

Determining Workcenter Lotsizes For Signal Kanbans 49

deterministic at workcenter 1 and is approximately 9 minutes

( 0. 15 hours) per container on either of the two (identical)

machines. The daily demand for end i terns X and Y and all

in-process items (A through L) as well as model cost parame-

ters are shown in Table 3-1. The setup times for all shop

workcenters as well as container processing times for all

items are shown in Table 3-2.

As a prelude to the analysis of the JIT system with signal

Kanbans, a simulation experiment was conducted using this

simulation model to attempt to ascertain if the shop would

work with standard Kanbans. (This required some adjustments

in the basic shop simulation model. ) As part of this simu-

lation analysis, the shop was run using the JIT system with

standard Kanbans and it did not work. Extremely large back-

orders resulted, -especially at workcenter l, that created

excessively long delays and demand not being met. After ex-

tended run time the shop effectively "collapsed."

Given that the JIT system with standard Kanbans will not

work with our example shop, a JIT system with signal Kanbans

appears as a workable alternative. In this modified JIT

system tne signal Kanban will be employed_in conjunction with_

workcenter 1 where large setup times are experienced. How-

ever, a crucial decision accompanying the use of signal

Kanbans is the lotsizes to be employed, and specifically in

the case of our example, the lotsizes at workcenter 1. As

such, the next portion of this chapter will be directed at

Determining Workcenter Lotsizes For Signal Kanbans 50

01' __.

Table 3-1. Cost and Demand Parameters for Example Shop

Item

A B c D E F H I J K L x y

Container Cost ($)

1500 1500 1500 1500 4125

375 375

3000 4500

375 375

9375 7975·

Daily Demand (Containers)

30 20 20 20 10 20 30 10 10 30 20 10 10

Holding cost = 25% per year of container cost Backorder cost = $5.63 per container per hour Setup cost = $65 per hour

Ul N

Table 3-2. Workcenter Processing and Setup Times.

Workcenter

1 2 3 4 5 6

Container Processinq Time (Hours) Setup Time (Hours)

.150

.135

.225

.170

.760

.760

.so

.01

.01

.01 N/A N/A

the development of an integer linear programming model for

determining the optimal lotsizes to be used in conjunction

with signal Kanbans.

3.3 AN INTEGER PROGRAMMING MODEL FOR DETERMINING LOTSIZES

3.3.1 THE MULTI-PRODUCT EOQ MODEL

Prior to actually developing the mathematical programming

model for determining lotsizes for signal Kanbans in a JIT

system the feasibility of using EOQ analysis for determining

lotsizes will be explored. A slightly modified version of

the following multi-product EOQ formula for single machines

with no backorders [S] will be used to determine the lotsizes

at workcenter 1 for two machines:

where,

Q. = J

cH. J

c. J

d. J

PT. J

R. J

j =

n =

R.

E CH. R.(1 - d.PT.) j=l J J J J

E CJ. j=l

is the holding cost for item j

is the order (setup) cost for item j

is the demand rate for item j

is the processing time per unit of item

is the annual demand for item j

item number (at workcenter l, j = A, B,

number of items processed at workcenter

Determining Workcenter Lotsizes For Signal Kanbans

j

c, D)

1.

53

\~ I "--

\ ~ \~

This particular EOQ formulation is employed in order to

eliminate backorders which, if allowed, would create delays

throughout the shop that used i terns produced at workcenter

1. The economic order quantity ( lotsizes) for each i tern

produced at workcenter 1 is, as follows: Q = 5£ containers A

and QB· = Q = Q · = 38 containers . . C D The signal. Kanbans used

in conjunction with each of the four i terns produced at

workcenter 1 are processed on a first-come, first-served ba-

sis.

The simulation model of the shop with these EOQ lotsizes

resulted in 40 total containers backordered after 21 weeks.

At this point in time with this number of backorders, the

shop was two full days behind schedule (for meeting demand)

which was viewed to be unacceptable. Thus, the determination

of signal Kanban lotsizes using the classical EOQ formula was

not felt to be a feasible procedure.

This was a result that could have been logically antic-

ipated. First, note that the EOQ formulation contains setup

costs but not setup times. Next let us define the

"workcenter production cycle" as the minimum time period re-

quired to produce the EOQ determined lotsize of each i tern

processed or assembled at the workcenter. At workcenter l,

the quickest way to produce the four i terns, A, B, C and D,

on the two available machines is to produce A on one machine

and then either B, C or D on the same machine. If we select

B such that the first machine produces A and then B, this

Determining Workcenter Lotsizes For Signal Kanbans 54 i

machine will require two setups per cycle and the time to

process A and B on machine 1 will be,

n * tmachine 1 = r (QJ. x processing time + setup time)

j=l

= (56)(.150) + (38)(.150) + 2(0.5)

= 15.1 hours

The processing time for items C and D on machine 2, computed

similarly, is tmachine 2 = 12.4 hours. However, demand for

item A (from Table 3-1) is 3.75 containers per hour. Since

QA, the EOQ determined lotsize, is 50 containers, the items

wi 11 be depleted in 13. 33 hours, which means that this

workcenter will never be able to meet demand and will always

be behind schedule.

Faced with such results, the natural inclination might be

to attempt to adjust the multi-product EOQ model to explic-

itly include feasibility considerations, such as setup times.

This is not as simple as it might initially appear since the

number of setups in a cycle will depend on the machine se-

lection rule and the particular sequence of jobs arriving at

each machine. The sequence of jobs, in turn, depends on

end-i tern demand and the particular synchronization of the

rest of the shop up to the present instant in time.

Rather than attempt to modify the multi-product EOQ model,

an integer mathematical programming model will be developed

in which the model constraints guarantee no backorders for a

Determining Workcenter Lotsizes For Signal Kanbans 55

workcenter employing signal Kanbans. The math programming

model wi 11 be developed using two alternative objective

functions, one which minimizes in-process inventory (re-

fleeting the Japanese philosophy of driving inventory levels I

to zero), and, an objective function that minimizes total

(i.e., inventory plus setup) costs.

3.3.2 MODEL CONSTRAINTS

When developing a model that will insure that no backer-

ders are incurred at workcenters in which signal Kanbans are

used, the shop workcenters may be "decoupled" so each can be

considered independently of the other. In a shop employing

the JIT system with one or more signal Kanban workcenters,

in general, it is not possible to decouple the workcenters

and consider each separately if any of the workcenters incurs

backorders. This is because under the JIT system, a

workcenter must wait for component parts from preceeding

workcenters, and, cannot begin processing until authorization

for outbound work-in-process is received. All workcenters

are integrally tied to each other and if one is delayed and

falls behind schedule, other workcenters may be affected.

However, it is our purpose to develop a mathematical pro-

gramming model with constraints that prohibit backorders al-

together. If there are no backorders in the entire shop and

in-process items (or raw materials) are available "in time"

Determining Workcenter Lotsizes For Signal Kanbans 56

------- - -- ----- --- -------- ------- -------- ------------------ ------------------- ---- ---- -----

at those workcenters needing them, any given workcenter will

have all of its needed pre-process i terns arriving "just-in-

time'' for processing, and, will have the outbound in-process

i terns ready 11 just-in-time 11 to be forwarded to workcenters

demanding these i terns. Consequently, all workcenters will

operate as though they are isolated from each other; pre-

process i terns will always be available when needed, and de-

mand for a workcenter' s processed i terns will simply be a

delayed image of the end-item demand. Therefore, constraint

equations that guarantee that a signal Kanban workcenter will

incur no backorders need not encompass variables that de-

scribe or define characteristics of any other workcenter as

long as we do not allow all other workcenters to have back-

orders. Thus, developing the constraint equations will be

greatly simplified since it will not be necessary to consider

any other workcenter except the one for which we are devel-

oping constraints.

There are three classes of constraints that must be sat-

isfied to insure that a workcenter with signal Kanbans will

have no backorders. First, the workcenter production cycle

time for each machine must be no smaller than the time it

takes to complete all work for that machine, i.e., the cycle

time for each machine must be greater than or equal to the

production time, including setups. This constraint is for-

mulated mathematically for a workcenter with signal Kanbans

as,

Determining Workcenter Lotsizes For Signal Kanbans 57

t. ~ l.

n l:

j=l ( q . . PT . + Y . . S )

l.J J l.J for i = 1 to m machines

where, n =number of total items to be produced at a

workcenter

qij = lotsize in containers for item j processed

on machine i

t. the production cycle time for the i th machine l.

at the signal Kanban workcenter

PTj = the processing time per unit of item j

Sj = the setup time (assumed to be the same for

all items at the workcenter in our example)

Y .. = l. J

1 if qij > 0

0 if q .. = 0 l. J

Strictly speaking, this equation only holds if lot pro-

duction is performed for more than one product on a machine.

If only one product is being processed on a machine, then the

product should be processed with standard Kanbans since no

setups are incurred.

The second constraint forces each i tern produced at a

workcenter to be produced on one machine; this will reduce

time-consuming and costly setups. That is, once it is de-

cided to produce a lot of an item on a machine, then all the

demand for that i tern during that machine's production cycle

Determining Workcenter Lotsizes For Signal Kanbans 58

-------"------ -- ------------ -----------------~----------~-------- ----------------- ----- ------------------- - -- -------

time will, be produced on that machine. This constraint is

written mathematically as follows:

m l: yij = 1

i=l

q .. l.J

where

S MY .. .. . l.J all q· ..

l.J

for j = 1 to n items

for all i and j,

are integer,

Yij is binary (as above), and

M is a very large positive constant.

The third and final constraint states that for each ma-

chine, demand for each i tern produced on that machine during

its production cycle must exactly equal the lotsize for that

item. This constraint is expressed mathematically as a pair

of constraints for each item and machine:

Q. s djti + (1 - Y. :)M J l. J

for all i and all j . Q. ;?:: d.t. - (l - yij)M J J l.

m where Qj = i!l qij (the sum of lotsizes for all

machines at a workcenter),

for all j,

and dj = the demand (containers) per unit time for

item j.

Consider the following two cases with respect to the above

two equations.

Determining Workcenter Lotsizes For Signal Kanbans 59

--~·

(l) if item j is produced on machine i, then q .. > 0 J. J

and hence Y = 1 . . ij

In this case, the equations may be rewritten as

Qj ::; d.t. J J.

Q. 2: d .t. / J J J.

which, taken together, are equivalent to

Q. = d.t .. J J J.

Therefore, if i tern j is produced on machine i, an equality

constraint holds.

(2) if item j is not produced on machine i,

0 and hence Y .. = 0. J. J

then q .. = J. J

In this case, the equations may be rewritten as

Qj Q.

J

::;

2:

d.t. + M J J.

d.t. - M. J J.

Since M is a huge positive constant, these constraints are

always trivially satisfied for any values of Q. d., and t .. J J J.

Therefore, if item j is not produced on machine i, there

is no third, final constraint to be satisfied.

When the three constraints presented in this section are

simultaneously satisfied, no backorders will occur at a sig-

nal Kanban workcenter.

3.3.3 THE INVENTORY MINIMIZATION MODEL

The constraints developed in the preceeding section, while

guaranteeing that demand will be met with no backorders, do

Determining Workcenter Lotsizes For Signal Kanbans 60

not necessarily result in a cost effective lotsize. There

are several objectives that can be employed that will result

in an effective or "optimal" lotsize, and in the following

discussion we will offer two of them, inventory minimization

and cost minimization.

The first objective to be examined is the minimization of

inventory which is consistent with the philosophy inherent

in the JIT technique to reduce inventory to the absolute

minimum level. The approach presented here is a generaliza-

tion of the approach reported by Monden [36] that is used by

Toyota. This objective function combined with our previously .

defined model constraints results in the following integer

mathematical programming model.

minimize Z =

subject to

t. ~ 1

m

n ! QJ.

j=l

n ! (qiJ.PTJ. + YiJ.S),

j=l for i = 1 to m.

l: yiJ" = 1, i=l

for j = 1 to n.

q .. :5 MY .. , for i = 1 to m and j = 1 to n 1J 1J Q. :5 d.t. + (1 - Y .. )M, J J 1 1J

for i = 1 to m and j 1 to n

Determining Workcenter Lotsizes For Signal Kanbans 61

--- --- --- - - ---- ---- - - - --- -- - -- --------- -~ - -- -- - ----~~-- -

Qj ~ djti - (1 - Yij)M,

for i = l to m and j = 1 to n

m Qj = z: q .. ,

i=l lJ for j = 1 to n.

Y .. is binary (0 or 1) lJ for i = 1 to m and j = 1 to

qij and Qj are integer

for i = 1 to m and j = 1 to

t. ~ 0 l

for i = 1 to m.

n

n

Recall the example shop we described previously with four

i terns produced at workcenter 1 (A, B, C and D) on two ma-

chines. Workcenter 1 was the only one of the six workcenters

in the shop that required the use of signal Kanbans; thus,

we will develop one integer programming model for this single

workcenter. For convenience and standardization of notation

we will redefine items A, B, C and D as 1, 2, 3 and 4 re-

spectively.

minimize Z = Q1 + Q2 + Q3 + Q4

subject to

4 Z: (0.1Sq1 J. + O.SOY1 J.)

j=l

4 Z: (0.1Sq2 J. + 0.SOY2 J.)

j=l

Determining Workcenter Lotsizes For Signal Kanbans 62

yll + y21 = 1

y12 + y22 = 1

yl3 + y23 = 1

yl4 + y24 = 1

'111 $ MYll

'112 $ MY12 '113 $ MY13 '114 s MY14 '121 s MY21 '122 s MY22 '123 $ MY23 '124 s MY24

Ql $ 3.75tl + (1 yll)M Ql ~ 3.75tl - (1 - yll)M

Ql $ 3.75t2 + ( 1 - y2l)M Ql ~ 3.75t2 - (1 - y2l)M

Q2 s 2.50t1 + ( 1 - yl2)M Q2 ~ 2.Sot1 - (1 - yl2)M

Q 2 $ 2.50t2 + ( 1 - y22)M Q2 ~ 2 .50t2 - (1 - y22)M

Q3 $ 2.50tl + ( 1 - Y13)M Q3 ~ 2.50t1 - (1 - Y13)M

Q3 $ 2. 50t2 + (1 - Y23)M Q3 ~ 2.50t2 - (1 - Y23)M

Q4 $ 2.50t1 + (1 - yl4)M Q4 ~ 2.50tl - (1 - yl4)M

Q4 $ 2.50t2 + (1 - Y24)M Q4 ~ 2. 50t2 - (1 - y24)M

Detennining Workcenter Lotsizes for Sign.al Kanbans 63

Ql = qll + q21 Q2 = ql2 + q22 Q3 = ql3 + q23 Q4 = ql4 + q24

Yij = 0 or 1 for i = 1 to 2 and j = 1 to 4

q .. , Q. integer for i = 1 to 2 and j = 1 to 4 J. J J

tl, t2 ~ 0.

The solution to this example model is,

Q1 = q 11 = 60 containers

q21 = 0

Q2 = q 12 = 40 containers

q22 = 0

Q3 = q 23 = 10 containers

q13 = 0

Q4 = q 24 = 10 containers

q14 = 0

t 1 = 16 hours

t 2 = 4 hours

Using our original example shop notation, this solution

indicates that the signal Kanban lotsize for i tern A at

workcenter 1 is 60 containers while the lotsize is 40 for B

and 10 containers each for C and D. Note that these lotsizes

are significantly higher than one might consider for a normal

Kanban workcenter. This solution also indicates that items

A and B will be processed on machine 1 with a production cycle

time of 16 hours for these two items. Items C and D will be

processed on machine 2 with a production cycle time of 4

Determining Workcenter Lotsizes For Signal Kanbans 64 -~-~--------------- - --------------~------~---------- ----~ -~ ---·--- ---

'

. i

I

hours. Machine 1 satisfies 2-days' demand during its two-day

cycle, and Machine 2 satisfies one-half day's demand during

its one-half day cycle.

3.3.4 THE COST MINIMIZATION MODEL

The mathematical programming model developed in the

preceeding section minimized inventory and ignored the direct

minimization of cost. This second version of the mathemat-

ical model will consider cost minimization as an objective

and is based upon a "rotation cycle policy" [23]. This model

encompasses both holding and setup cost in the determination

* of an optimal Q .. The objective function now becomes J

This function stipulates that the sum of holding and setup

costs for all items at the workcenter is minimized. To this

new objective function we add all the constraints developed

to prevent backorders for the previous mathematical program-

ming model that minimized inventory.

The optimal solution to this model for our example shop

is QA= 60, QB= 40 (both processed on machine 1), QC= QD =

38 containers (both processed on machine 2). Machine 1' s

production cycle time is 16 hours and machine 2' s is 15. 2

hours.

Determining Workcenter Lotsizes For Signal Kanbans 65

The two mathematical programming models were subsequently

solved for a variety of different setup times at workcenter

1. The different lotsizes obtained from these setup times

for workcenter 1 are shown graphically in Figure 3-2.

The results in Figure 3-2 for our example shop show that

the minimum inventory integer programming model generates·

consistently lower lotsizes than the minimum cost model for

all setup times. This is not unexpected since the objective

of the minimum inventory model is to minimize lotsizes. In

order to analyze the implications of these models on shop

costs it is necessary to employ the simulation model devel-

oped previously.

3.4 SIMULATION ANALYSIS OF MODEL LOTSIZES

Recall that total cost in our shop is the sum of inven-

tory, backorder and setup costs. Since both mathematical

programming models eliminate backorders, total cost is simply

the sum of inventory and setup costs. When the various

lotsizes shown in Figure 3-2 are substituted into the simu-

lation model of the example shop the total cost values at

workcenter 1 shown in Figure 3-3 are generated. Notice that

while the "minimum cost" model produced the higher lotsizes

in Figure 3-2 this same model resulted in the lower cost in

Figure 3-3. This result occurs because the larger lotsizes

Determining Workcenter Lotsizes For Signal Kanbans 66

300 en a: L&J z, ...... --:...-c:r··· ... 250 z 0 (.)

z - 200 LaJ N -en t-Q .;J 150 L&J ·~ < <.:> L&J 100 a:: " (!) <[

50

.50 .75 1.0

SETUP TIME (in hours)

Figure 3-2. The Effect of Setup Time on Lotsize.

67

------------------- -----------

120

min inventory

100 I Standord Konban

~ $84,247 0 I . 0 0 • .. 80 - min cost -c: -t-VJ 0 60 0 _J

~ $49,129 0 40 t-

20

.25 ·.50 .75 1.0

SETUP TIME (in hours)

Figure 3-3. The Effect of Setup Ti~e on Total Cost.

68

- --· --- -- ------ --- ------------------.. --·

generated by the "minimum cost" model reduce the number of

setups, and in this particular example shop, costs are very

sensitive to setups.

While it is inappropriate to develop any conclusions from

only one data point (i.e., the example shop) these results

do have several implications for the interested manager con-

templating the installation of a JIT system with signal

Kanbans. Al though the Japanese philosophy inherent in the

JIT system emphasizes the minimization of inventory ( fre ...

quently exclusive of cost considerations), it may be prudent

for the manager to consider cost more prominently. From

Monden' s book [36, pp. 215-216] it appears that Toyota em-

ploys the minimum inventory model approach we have presented

in this study. However, the results presented in Figure 3-3 I

indicate that a minimum cost approach is superior, at least

for this particular example. In other words, there may be

some penalty associated with a philosophy that focuses on

inventory reduction without at least a glance at the cost

implications. In order to explore this question further we

will again employ our simulation model to analyze the impact

of container processing times on the example shop.

3.5 SIMULATION ANALYSIS OF CONTAINER PROCESSING TIMES

In this experiment we will explore the impact of varying

container processing times (with constant setup times) on

Determining Workcenter Lotsizes For Signal Kanbans 69

------------- ___________ / ______________________ --- ------- ------ ----- ---- --- -- -- --- - - -- --- .

---- ~--------~-------~ _ ___,;

workcenter lotsizes and costs. First, various lotsizes are

generated from the two integer programming models. These

results are graphically displayed in Figure 3-4. Notice from

Figure 3-4 that, as in Figure 3-2, lotsizes for the minimum

inventory model are consistently smaller than those for the

minimum cost model when container processing times are varied

for our example shop. It is interesting to note in Figure

3-4 the rather dramatic impact a slight reduction in con-

tainer processing times has on lotsizes. At least for this

example shop, for container processing times greater than

approximately 0. 15 hours, a slight decrease in processing

times results in a very large decrease in lotsizes.

Employing the lotsizes obtained from the two integer pro-

gramming models (with different container processing times)

in the simulation model of the example shop generates the

cost results shown graphically in Figure 3- 5. The minimum

cost model dominates the minimum inventory model over the

range of container processing times explored for our example.

It is interesting to note that contrary to expectations,

for the "minimum inventory" model, costs increase as proc-

es sing times decrease. The explanation for this occurrence

is that, as container processing times are reduced, smaller

lotsizes result which creates more setups. Thus, the cost

curve shown for the "minimum inventory" model in Figure 3-5

is a result of high setup costs in our shop. With the cost

parameters used in this shop example, the increased cost due

Determining Workcenter Lotsizes For Signal Kanbans 70

----------- ---------~- - ----~-----~ ---- -- ------~-~--- ------- ----- --- - -- -- ------

'

--- ------. -------

350

300 (/) a:: w z -<( .- 250 z 8 z

200 l.&J N -(/) ... 9 150 LaJ

~ (,:> LaJ

100 a:: <.:> (,:> <(

50

.13 .14 .15

CONTAINER PROCESSING Tl ME (in hours)

Fiqure 3-4. The.Effect of Container Processinq Time on Lotsize.

71

140 .,...... 0 120 0 0 .. -- 100 c

~ -I- 80

miJcost (/) 0 0 60 _J <( 40 b I- 20

.13 .14 .15 .16

CONTAINER PROCESSING TIME (in hours)

Figure 3-5. The Effect of Container Processing Time on ~otal Cost.

to setups is much greater than the cost savings for reducing

inventory. Although this result is example dependent, it

still indicates that it may be important for a manager to

consider the cost implications of signal Kanban lotsizes

prior to implementing a JlT system.

3.5.1 COMPARING SIGNAL AND STANDARD KANSAN SYSTEMS

In our final simulation model experiment we will compare

the JI'i' system employed in our example shop using standard

Kanbans and signal Kanbans at workcenter 1.

In order to operate the example shop using standard

Kanbans, setup times at workcenter 1 must be . 025 hours or

less (a value determined by simulation model experimenta-

tion). The total cost at workcenter 1 for the shop with

standard Kanbans (and setup times of .025 hours) is $84,247,

a point designated in E'iglire 3-3. However, notice in Figure

3-3 that the total cost incurred at workcenter 1 using signal

Kanbans with this same setup time is significantly lower

($49,129). The total cost with signal Kanbans at workcenter

1 increases as setup times increase, but remains below the

standard· Kanban value up unti 1 setup times of about 0. 5

hours.

Al though this result is based on just a single example,

it does have significant implications. There is a tendency

among managers implementing a JIT system to attempt to reduce

Determining Workcenter Lotsizes For Signal Kanbans 73

I~

!

setup times to the point where a standard Kanban system can

be used. However, observing Figure 3-3, it can be seen that,

at least for this examp1e shop, if setup times are reduced

to a point where standard Kanbans are feasible then signal

Kanbans may be even more cost efficient than standard

Kanbans. As such, it might be that .a standard Kanban system

should not always be the ultimate objective, and, that a

signal Kanban system should be retained as setup times are

reduced. Carrying this logic one step further, under certain

circumstances and. operational shop scenarios a JIT system

with signal Kanbans at all workcenters might be more cost

effective :than a JIT system with standard Kanbans. At the

very least, these results indicate that the implementation

of a JIT system encompasses a number of interrelated and

complex considerations that should be ful.ly explored before

the final system is operationalized.

3.6 SUMMARY

The purpose of this chapter has been to develop a model

for determining the appropriate lotsizes to use in conjunc-

tion with signal Kanbans in a shop using the Just-in-Time

technique with Kanbans. It was demonstrated via a simulation

model of an example shop that the classical multi-product EOQ

model does not always work in a JIT shop. As a result, two

integer mathematical programming models for determining sig-

Determining Workcenter Lotsizes For Signal Kanbans 74

na1 Kanban lotsizes and eliminating backorders were devel-

oped. One model version employed an objective function that

minimized inventory while the other model version minimized

cost. The impact of these results on total workcenter costs

was determined ·using the simulation model of the example

shop. These simulation results, while not providing any

universal guidelines or conclusions, do offer several impli-

cations for the manager considering the implementation of a

JIT system. In general, these implications are that it may

be prudent to consider inventory and setup costs rather than

simply seeking to reduce inventory to its minimum level in a

JIT system; and, under certain conditions a signal Kanban

system may be more cost effective than a feasible standard

Kanban system.

The next chapter focuses on determining for an American

environment which factors influence the number of Kanbans at

a workcenter and how the initial number of Kanbans should be

determined in such a setting.

Determining Workcenter Lotsizes For Signal Kanbans 75

4.0 AN ANALYSIS OF FACTORS INFLUENCING THE NUMBER OF

KANSANS REQUIRED AT A WORKCENTER

The purpose of this chapter is to analyze two aspects of

the JIT technique that confront a production manager imple-

menting the JIT technique for the first time in a uniquely

American production ·environment. First, the factors that

influence the number of Kanbans required at workcenters in a

JIT system wi 11 be identified and discussed. The impact of

these factors on the efficiency of a JIT system will be dem-

onstrated via a simulation model of an example production

operation. Second, a methodology will be presented for de-

termining the initial number of Kanbans to use in a JIT sys-

tern. This methodology will also be demonstrated via a

simulation model of a production system that does not gener-

ally exhibit the characteristics of a Japanese production

operation.

4. 1 A JAPANESE APPROACH TO DETERMINING THE NUMBER OF

KANSANS

Prior to analyzing the factors that influence the number

of Kanbans used in a JIT system it will be beneficial to ob-

serve how the Japanese set the number of Kanbans at a

An Analysis of Factors" Influencing the Number of Kanbans Required at a Workcenter 76

workcenter. The Toyota Motor Company is one Japanese firm

that has had particular success using the JIT system; thus,

we will use their procedure for setting the number of

Kanbans, as follows [36]:

Number of C: (demand)(lead time)(l + safety factor) (4-1) Kanbans container capacity

where lead time =processing time +waiting time + conveyance time + Kanban collecting time [36, p. 70]

Kanban collection time in this formula is the time during

which Kanbans are waiting to be picked up or returned to the

beginning of a production operation.

If the safety factor in (4-1) is set equal to zero and if

demand is expressed in containers per unit time then the

formula becomes:

Number of Kanbans ~ (demand)(lead time) (4-2)

Thus, the number of Kanban~ (according to the Toyota Com-

pany) is at least the lead time demand expressed in terms of

containers. This formulation has intuitive appeal in that

it allows only enough buffering to compensate for lead time

demand. We will employ this formulation in the next .section

of the chapter as a basis for determining the factors that

influence the number of Kanbans, given significant variabil-

ity in the production system.

The Japanese have significantly reduced the variability

in their production system. The slight variability in the

system that does exist is compensated for by the use of the

An Analysis of Factors Influencing the Number of Kanbans Required at a Workcenter 77

safety factor in equation (4-1), which is typically deter-

mined by the workcenter foreman. The Japanese can employ

such a simplified and direct approach because of the fore-

man's experience combined with the small amount of variation

in the production parameters.

However, if· the production system is not under such rigid

control then setting a safety factor or estimating lead time

or lead time demand becomes much more complicated, and, the

straightforward Japanese approach for determining the re-

quired number of Kanbans becomes less effective and predic-

tive. In the following section we will employ the Japanese

formulation for determining Kanbans shown in (4-2) as a basis

to examine the effect of stochastic variation on a JIT sys-

tern, and, identify the factors that influence the number of

Kanbans to use in a system with variation.

4.2 FACTORS INFLUENCING THE NUMBER OF KANSANS TO USE IN A

JIT SYSTEM WITH VARIATION

In order to identify the factors that influence the number

of Kanbans to use at a workcenter we will take a simplified

approach and look only at a single workcenter using a single

Kanban (and container) during one time period. Furthermore,

we will assume that the workcenter encompasses only one ma-

chine; the shop produces only one product; that waiting time,

An Analysis of Factors Influencing the Number of Kanbans Required at a Workcenter 78

conveyance time and Kanban collecting time are all zero or

negligible relative to processing time; that product demand

arrives instantaneously during each cycle period; and, setup

times are zero and all processing times are equal. Given

this sample scenario we will attempt to determine the factors

that create the likelihood of a backorder at the workcenter

since it is backorders that necessitate additional Kanbans

to use as a buff er unti 1 the backordered demand can be met

from excess capacity in succeeding periods.

In order to facilitate understanding of the forthcoming

discussion it is necessary to define several terms, as fol-

lows. Container demand cycle (CDC) is the interarrival time

of orders for the shop's final product; therefore, CDC =

l/demand. Cycle processing time (CPT) is the time required

to complete production at a workcenter during one container

demand cycle assuming the workcenter is idle when the cycle's

orders arrive. (For example, if an order for 3 identical

component parts arrives each CDC at a workcenter, and if no

setups are necessary and the part processing time is 19 min-

utes, then the CPT equals 57 minutes.) Cycle throughput

velocity (v) is the average number of items necessary to meet

demand per machine per workcenter per time period. (For ex-

ample, if workcenters A and B each have one machine and if 3

items are demanded at workcenter A and 10 items at workcenter

An Analysis of Factors Influencing the Number of Kanbans Required at a Workcenter 79

B per hour, then v A equals 3 i terns per machine per hour and

vB equals 10 items per machine per hour.)

Now, recall formula ( 4-2) that was presented earlier to

determine the number of Kanbans:

Number of Kanbans 2: (demand)(lead time) {4-2)

Given our previous assumption that our example workcenter

requires only one Kanban, (4-2) can be rewritten as,

l/demand 2: lead time (4-3)

However, also recall that we assumed that waiting time,

conveyance time and Kanban collecting time are all equal to

zero or negligible, which means that lead time equals proc-

essing time. Thus (4-3) can be reformulated as,

l/demand :.' CDC ~ lead time =. CPT (4-4)

or,

CDC 2: CPT. (4-5)

The cycle processing time (CPT) can also be expressed as,

CPT = t(µPT + EPT) (4-6)

where µPT = the mean processing time per item

tPT = the random error in the processing time.

If processing times are assumed to be independent and

identically distributed, the mean and standard deviation of

the CPT can be written as follows:

µCPT = vµPT

°CPT = vl/2 0 PT"

An Analysis of Factors Influencing the Number of :Kanbans Required at a Workcenter 80

___ _,_ - - - -------------~-----------

Employing these formulas, the following statements can be

written describing the probabi1ity that CDC will equal CPT:

(4-7)

or,

(4-8)

where ZP is the number of standard deviations that the ·cPT

must be greater than its mean in order for CDC to equal CPT.

As an example, if CDC = one hour, µCPT = O. 85 hour and

aCPT = 0~1786 hours then ZP =0.84. If it is assumed that the

CPT is normally distributed, then the probability that the

CDC will be exceeded is 0. 201. In other words, the proba-

bility that• backorder will occur (i.e., there is an insuf-

ficient number of Kanbans or containers) is 0.201.

Equation (4-8) can also be reformulated in terms of vari-

ous production parameters, as follows:

utilization ~ util = vµPT/CDC

coefficient of variation ~ CV = aPT/µPT

(4-9)

(4-10)

µCPT = vµPT = (vµPT/CDC)(CDC) = (util)(CDC) (4-11)

Thus, the standard deviation of CPT can now be written as

- 1/2 °CPT - 0 PTV

An Analysis of Factors Influencing the Number of Kanbans Required at a Workcenter 81

-------- --- ---- ----·

' = (CV)(util)(CDC)/v112 , v # 0. (4-12)

Recall from (4-7) that,

CDC = µCPT + 2P°CPT

which, with (4-11) and (4-12) substituted, becomes

CDC = (util) (CDC) + Zp(CV) (util) (CDC/v112 ) (4-13)

or,

zp = (v112;cv)[(l/util) - l], O < util s 1

and CV # 0

(4-14)

(A coefficient of variation (CV) equal to zero implies no

probability of a backorder.)

From equation (4-14) we can now identify the factors that

influence the probability of backorders, and hence, the num-

ber of Kanbans needed. The first factor is the throughput

velocity, v. Zp varies directly with the root of the

throughput velocity; therefore the probability of a backorder

will vary inversely with the throughput velocity. As the

throughput velocity decreases, the probability of a backorder

will increase, as will the need for additional Kanbans.

A second factor from (4-14) that influences the number of

Kanbans needed is the coefficient of variation (CV) of the

processing time at the workcenter. Zp varies inversely with

the coefficient of variation, therefore, the probability of

backorders will increase as CV increases. Further, the

An Analysis of Factors Influencing the Number of Kanbans Required at a Workcenter 82

probability of backorders will increase faster for CV in-

creases then it wi 11 decrease for throughput velocity in-

creases.

The third factor identified from (4-14) is the utilization

of machines at the workcenter. As the utilization rate in-

creases, ZP will decrease, and thus, the probability of

backorders and the number of Kanbans will increase.

A fourth factor that can also influence the number of

Kanbans employed at a workcenter is whether or not

autocorrelated processing times exist. Recall that in the

above derivation, processing times were assumed to be inde-

pendent and identically distributed. If n items are produced

per CDC at a workcenter and the processing times are posi-

ti vely autocorrelated so as to follow an AR( 1) process (see

Box and Jenkins [3]) rather than being independent, then,

where,

I

0 CPT = OCPT(l + on) (4-15)

o CPT = the standard deviation of cycle processing

time with autocorrelation following an AR( 1)

process.

oCPT = the standard deviation of cycle processing time

with independent and identically distributed

processing times

o = a function strictly greater than 0 for all n. n

An Analysis of Factors Influencing the Number of Kanbans Required at a Workcenter 83

In the case of positive autocorrelation, equation { 4-14)

becomes,

z p (4-16)

where Z has the same p meaning as but for the

autocorrelated case. Thus, if processing times are

autocorrelated, Z p wi 11 decrease by a factor of ( 1 + 1 n)

from the case with no correlation, and, the probability that

the CDC will be exceeded and there will be backorders in-

creases.

4.2.1 A SIMULATION CASE EXAMPLE

In order to demonstrate the impact of these factors on the

number of Kanbans used at a workcenter, an example shop will

be simulated with varying operating conditions. The example

shop is described in Chapter 2. The product demand is 200

uni ts for X and 100 uni ts for Y. The container size is 20

units for X and 10 units for Y.

In order to provide a more realistic test of the factors

we identified via equation (4-14), the assumptions originally

specified prior to the development of equations ( 4-1) through

(4-14) will be relaxed. In our example shop all workcenters

are linked together and simulated for a number of cycles, two

final products are produced, waiting times are not zero and

An Analysis of Factors Influencing the Number of Kanbans Required at a Workcenter 84

-------· ------- -------

the arrival of demand during each cycle i$ not instantaneous.

Processing times are normally distributed and assumed to

follow an AR( 1) series with a positive autocorrelation of

0. 5. The means of the container processing times for each

product at each workcenter and the setup times at each

workcenter are shown in Table 4-1.

The example shop was simulated and the results are. shown I

in Table 4-2. Table 4-2 contains the value of Z p' computed

for each workcenter using equation ( 4-16), with an ordinal I

ranking of the workcenter' s Z 11 score 11 the coefficient of p I

variation (CV), the utilization rate (util), the cycle

throughput vel.oci ty (v), the number of Kanbans determined

from trial and error, and, the average number 0:5 backorders

incurred at the workcenter. (The average number of backer-

ders at a workcenter is the average number of withdrawal

Kanbans which cannot be matched with an item of finished in-

ventory.)

The objective of this simulation analysis is primarily to

determine if the methodology used to determine the probabil-

i ty of backorders (and, thus, if there are enough Kanbans),

developed in the previous section, is effective given a more

realistic and complex production environment than the one

used to develop equations (4-14) and (4-16). In our example, I

the value of Z was first computed for each workcenter using p the indicated values for CV, util and v, and then ranked

An Analysis of Factors Influencing the Number of Kanbans Required at a Workcenter 85

------------------ - - ---- --------. --- ----- ------------- - -- ---- -- ----- ----- --- - --- --- --

Table 4-1. Parameters for the Simulation Example.

No. of Container Processing Setup Time Workcanter Machines Time (Hours) (Hours)

"x l1y

1 1 0.124 0.062 0.005

2 1 0. 138 0.069 0.005

3 1 0.328 0.164 0.020

4 1 0. 1768 0.0884 0.012

s .. 0.648 N/A N/A I

6 1 N/A 0. 736 N/A.

Table 4-2. Simulation Results with Autocorrelated Processing Times.

Fae ton Influencing &he IUllbu 1C ~1ob101 lllHllHhm BHUIU

l Avenge Hu.bar or rarror•anca

Workcenter CV utl I v l rank Backordara kanbans Rank

• .J .126 9 2. ll 1 • 0Jla2 • 1

.. .l .122 6 2.056 2 .061l ' 2

5 .l .199 • .190 l .0019 2 l

2 .l .189 10 .800 .. .0,11 2 .. J .J .920 .. .lH 5 .1060 2 5 6 • .1 .911 ' .291 6 ... 101 l 6

ordinally from 1 to 6 with the best performance being ranked

first. These values and rankings are shown in the first part

of Table 4-2. Note that a high value for Z indicates a low p

probability of backorders. For example, in Table 4-2, a I

value of 2. 33 for Z for workcenter 1 indicates a lower p probability of a backorder than a Z value of 2. 056 for p

workcenter 4; hence, workcenter 1 is ranked higher than

workcenter 4. These Z values (and the rankings) were all p

determined manually using equation (4-16).

Alternatively, the second part of Table 4-2 consists of

the results obtained using the simulation model. The number

of Kanbans used per workcenter was determined by trial and

error with the basic objective of reducing backorders to an

average of .5 or less per hour. The performance ranking for

the simulation results was the number of Kanbans required per

workcenter to meet this arbitrarily selected guideline.

Thus, workcenters 1 and 4, which each requires only one

Kanban, are ranked first and second. The differential rank-

ing between these two workcenters is based on fewer average

backorders at workcenter 1 than workcenter 4. The differen-

tial rankings for workcenters 5, 2 and 3, each of which re-

quires two Kanbans is determined the same way.

The significant aspect of the results in Table 4-2 is that

the perf ortnance rankings of the workcenters determined manu-I

ally using the Z formulation and with the simulation model, p

An Analysis of Factors Influencing the Number of Kanbans Required at a Workcenter 88

are in complete agreement. In other words, the use of Z p

appears to be an entirely effective means for analyzing the

influence of the production factors noted on the number of

Kanbans required. These results also appear to verify that

the coefficient of variation (CV), utilization (util) and

throughput velocity (v) directly influence the number of

Kanbans required at a workcenter.

4.3 A SIMULATION APPROACH FOR DETERMINING THE INITIAL

NUMBER OF KANSANS AT A WORKCENTER

The first part of this chapter consisted of an analysis

of the factors in a production system that will influence the

number of Kanbans that must be used at a workcenter. In this

second part of the chapter a methodology using simulation

will be demonstrated for determining the number of Kanbans

to use at a workcenter given a realistic and variable pro-

duction environment. Such a production environment would

include multiple machines at workcenters, stochastic proc-

essing times and setup times, and, complex i tern sequences

through the production system.

As a prelude to the development of the methodology for

determining the number of Kanbans at a workcenter we will

make the simplifying but important assumption that the pro-

duction system will operate with no backorders. That is, it

An Analysis of Factors Influencing the Number of Kanbans Required at a Workcenter 89

will be assumed that the objective is to specify just enough

Kanbans throughout the shop so that backorders will never be

incurred at any workcenter. This is a reasonable objective

since systems using the JIT technique are typically very

sensitive to backorders, especially in large shops where many

workcenters are cascaded. This assumption will enable all

workcenters to be "decoupled" from each other since the items

produced at each workcenter will always be available at other

workcenters when they are needed.

Recall the previous formulation (4-2) for determining the

number of Kanbans at a workcenter used by the Toyota Company:

Number of Kanbans ~ (demand)(lead time) (4-2)

We will employ basically the same formulation for deter-

mining the number of Kanbans, as follows:

(4-17)

where, n = the number of sets of Kanbans for an i tern at a

workcenter

D =the maximum demand for that item's final product in m containers

L0 . 95 = the maximum lead time at that workcenter determined

to be the 95th lead time value from 100 simulated

lead times ranked in order of magnitude from lowest

to highest. (Lead time is determined in this man-

ner because it was concluded from experimentation

that the absolute maximum lead time allowed too

An Analysis of Factors Influencing the Number of Kanbans Required at a Workcenter 90

many extra Kanbans floating in the system whereas

the 95th highest lead time did not.)

(Note that in ( 4-17) "n" equals the number of sets of

Kanbans. The total number of actual Kanbans is computed

by multiplying the sets of Kanbans by the number of contain-

ers of an item that are needed to make a container of final

product. For example, assume at a workcenter that two sets

of Kanbans are necessary for product X as determined by

equation (4-17). If at the workcenter 2 B's and 3 A's are

required to produce an X then the total number of actual

Kanbans needed is 4 for Band 6 for A or 10 overall.)

The essential difference between (4-2) and (4-17) is that

our latter formulation a.ssumes a production operation wherein

the variability in production parameters makes it difficult

to estimate Dm and L. Alternatively, in the first (Japanese)

formulation the lesser degree of variability would result in

relatively accurate estimates of demand and lead time. As a

result of these differences, we will employ a simulation ap-

preach to determine the parameters on which the number of

Kanbans is based in (4-17).

As already indicated, the maximum forecasted demand is

emplo¥ed as the estimate for Dm This is somewhat different

from the Japanese approach where actual demand is simply es-

timated and multipled by one plus a safety factor set by the

foreman as shown in (4-1). The determination of lead time

An Analysis of Factors Influencing the Number of Kanbans Required at a Workcenter 91

( L0 . 95 in equation ( 4-17)) is generally more complex. In

some instances it may be possible to use analytical results

for time in system to determine lead time if the workcenter

can be represented by a simple queueing model such as an

M/M/l or M/M/c model. However, it is more likely that lead

time cannot be determined analytically in which case a simu-

lation approach is required.

4.3. 1 THE SIMULATION MODEL

The simulation methodology for determining lead time will

be described via an example which is a slightly modified

version of the production operation described earlier. Demand

for X and Y will be changed to 190 and 100 uni ts per day,

respectively, in order to cofuplicate the system by throwing

demand out of synchron'ization. The setup and processing

times as well as the number of machines at each workcenter

are shown in Table 4-3. Notice that the processing times for

the containers of product X i terns are twice as large as the

containers for i terns used for product Y at workcenters l

through 4 since because X has a container size of 20 uni ts

whereas Y has a container size of 10 units.

The simulation methodology will be demonstrated via two

examples, one for workcenter 6 and one for workcenter 4. We

will begin with workcenter 6 because it is relatively easy

An Analysis of Factors Influencing the Number of Kanbans Required at a Workcenter 92

Table 4-3. Simulation Example Para.meters.

No. of Container Processing Setup Time Workcanter Machines Time (Hou rs) (Hours)

"x lly

1 2 o. 166 0.083 0.007

2 2 0.184 0.092 0.007

3 2 0.452 0.226 0.028

4 2 0.250 0.125 0.017

5 1 0.682 NIA NIA

6 1 NIA 0.736 NIA

to model since it has only one demand stream. In other words,

workcenter 6 satisfies demand for product Y only by assembl-

ing parts into 10 containers of Y every eight-hour day. As

a result, container demand is deterministic with an interar-

rival time of 0. 8 hours. Assume service times follow a

normal distribution and recall that there is only one machine

at the workcenter.

If we assume that there is sufficient capacity and there

is a sufficient number of Kanbans to prevent backorders at

each workcenter, then workcenter 6 will never experience a

shortage of in-process i terns. These assumptions enable us

to decouple workcenter 6 from the rest of the operation as

discussed above and consider it as ari isolated case with de-

mand that has a deterministic interarrival pattern and a

normal service distribution with a single server, which is

actually the description of a D/N/l queueing model.

In order to obtain an estimate for lead time, a D/N/l

queue is simulated. Time in the system, which is our esti-

mate of lead time, is recorded for every fiftieth arrival.

(The fiftieth arrival is collected because the lead times are

autocorrelated in the sense that if one job has a long lead

time the probability increases that the next job will have a

longer than normal lead time. By collecting only the fifti-

eth observation a sequence of 11 independen-::. observations is

obtained.) 100 observations -are obtained in this manner.

An Analysis of Factors Influencing the Number of Kanbans Required at a Workcenter 94

/

These 100 lead time observations are ranked from smallest to

largest and the 95th highest value is used as our estimate

of lead time as explained previously. This value is then

used (together with an estimate of maximum forecasted demand,

D ) to compute the number of Kanbans required at workcenter m 6 according to equation (4-17).

The second simulation example for workcenter 4 is more

complex than workcenter 6 since workcenter 4 must process

components to satisfy demand at two workcenters, 5 and 6.

Demand from workcenter 5 for component items to tnake X (i.e.,

2 F's and 1 J) arrives at the same rate as the demand for

end-item X (i.e., 190/20 = 9.5 containers) due to the "pull"

nature of the JIT shop described earlier. Therefore, the

interarrival time for orders at workcenter 4 from workcenter

~is 0.842 hours (i.e., 8 hours/9.5 containers). Similarly,

demand from workcenter 6 for component i terns to produce Y

(i.e., 2 L's and 1 M) arrives every 0.80 hours at workcenter

4. Thus, the demand for orders arriving at workcenter 4 is

actually a superposition of two demand streams. There are

also 2 machines at workcenter 4.

If -sufficient Kanbans are provided to always meet demand

at workcenters 5 and 6, then workcenter 4 may be decoupled

from them. Likewise, if workcenter 1, 2 and 3 also have

sufficient Kanbans to always meet demand, then in-process

items will always be available at workcenter 4. This enables

An Analysis of Factors Influencing the Number of Kanbans Required at a Workcenter 95

us to consider workcenter 4 as an isolated case and model it

separately from the remainder of the shop just as we did with

workcenter 6.

Workcenter 4 can be treated as a queueing model with

interarrivals following the superposition of two

deterministic rates, normal service times and two servers.

This model was simulated until 100 independent observations

could be collected with the 95th highest observation selected

as the estimate for lead time. This estimate for lead time

was subsequently used in equation ( 4-17) to determine the

number of Kanbans required at workcenter 4.

The simulation methodology described for workcenters 4 and

6 was employed for all six workcenters in the production

system described in Figure 2-1. Estimates for L and n were

obtained for three different cases of variation in processing

times: CV= 0.1, 0.2 and 0.3. These results are shown in

Table 4-4.

Next, a simulation model of the example production system

described in Figures 2-1 and 2-2 and Table 4-3 was simulated

using the number of Kanbans determined by the methodology

described above and shown in Table 4-4. Processing times at

each workcenter were assumed to be independent and iden-

tically distributed with CV= 0.1, 0.2 and 0.3, in that or-

der. The simulated production operation ran with all demand

being met and no backorders for all three cases. This indi-

An Analysis of Factors Influencing the Number of Kanbans Required at a Workcenter 96

lO ........

Table 4-4. Number of Kanban Sets for Processing Times Which are Independent and Identically Distributed (IID).

. Coeffjcient of Lead time Number of Kanban Sets Variation for

Workcenter L nx "v Processing Times

1 0.61 1 1 2 0.75 1 1 3 o. 78 1 1 CV = 0. 1 4 0.60 1 T 5 0.78 1 N/A 6 0.90 N/A 2·

1 0.64 1 1 2 0.74 1 1 3 0.87 2 2 CV = 0.2 4 0.62 1 t 5 0.97 2 NIA 6 1.23 N/A ·2

1 0.65 1 1 2 0.72 l 1 3 0.88 2 2 CV = 0.3 4 0.61 1 1 5 1. 14 2 N/A 6 2.17 NIA 3

cates that, at least for this example production system, the

methodology for determining the number of Kanbans is correct.

In order to determine if there were too many Kanbans in

the system, i.e., too much "slack," sensitivity analysis was

conducted using the simulation model. This analysis con- I I

·! sisted of removing a set of Kanbans for an i tern at a

workcenter that had more than one set of Kanbans and repeat-

ing the simulation. Results indicated that backorders oc-

curred at each workcenter when Kanbans were removed.

Similar simulation model results were obtained for a sec-

ond simulation analysis wherein processing times were assumed

to be positively autocorrelated.

4.3.2 THE EFFECT OF LESS-THAN-IDEAL PRODUCTION FACTORS

This methodology can be used to explore the effect of

less-than-ideal production factors on the number of Kanbans.

Consequently, as a final simulation analysis, the influence

of the various factors identified in the first part of this

chapter on the number of Kanbans used was examined within the

context of this example production system and simulation

model. Figures 4-1, 4-2 and 4-3, indicate re spec ti vely the

effects of the coefficient of variation (CV), utilization

(util) and positive autocorrelation on the number of Kanbans.

An Analysis of Factors Influencing the .Number of Kanbans Required at a Workcenter 98

6,__ ______________ ...,... ______ _,.. ______ _,

7

6

--en (I) ... 5 ~ 4 z 0 <· .c: CD .._... ~

z 0 < "' 4 ~ (j) 3 p7.75 LL. UJ 92.0/o utilization 0 :E cc - L&J I- 3 CD Q 2 < 2. ::::> ~ z

2.

0 .·1 .2. .3 COEFFICIENT OF VARIATION (CV)

Figure 4-L The Effect of Variability in Processing Time on the Number of Kanbans at a Workcenter.

99

---~------ -- ---- -- - -- -- - --------·-- --- --- --- -----

a 10

-/ 9 -en a ~ en :::i 0 ES z -<= 7 <( ~

~· . Cil

0 6 z It) <( en ~

LU 4 5 lJ... 2 0 - c::: .... 4 w c aJ <( 3 :? ~ 2 crPT=.276 :::::>

p=.75 2 z

eo es 90 95 UT1LJZAT10N (0/o)

Figure 4-2. The Effect of Utilization on tl:.e Number of Kanbans at a Workcenter.

lCO

6..--------.-------_,.. ______ .......,. ______ __

7

5

....... (Jj en 4 5 '"'" z ::s

0 < .s:: CD ......... z ~ 4

<( 0 ~ ~ O') 3 LL LL.I 0 .~ <:>l=.3 er r- 3 LL.J

92.0/o utilization CD c 2 ::? <( ::> ~ z

2

0 .25 .5 .75 AUTOCORREJ ATlON { p )

1'igure 4-3. The Effect of Correlation of Processing Times on the Number of Kanbans at a Workcenter.

101

----------- - - ---------- ---- ---- ----- -- -------~----------------

The curves shown in these figures further demonstrate the

effects predicted by equations (4-14) and (4-16).

For the production manager contemplating the implementa-

tion of the JIT technique with Kanbans, the type of analysis

embodied in Figures 4-1 ,4-2 and 4-3 can have a significant

impact. For example, Figure 4-1 indicates that if the vari-

ability in processing times can be reduced, and processing

times standardized, then the number of Kanbans can be re-

duced. This is a factor that has enabled the Japanese to

employ the JIT technique successfully. Alternatively, if a

manager is considering use of the JIT technique in a job-shop

type production operation, where it is difficult if not im-

possible to reduce the variability in processing times, then

the number of Kanbans required may be so high as to render

the JIT system ineffective.

Similar insight into the JIT technique can be gleaned from

Figure 4-2. If new, more efficient machines, more effective

production processes and better trained workers can be

achieved which will reduce utilization, then the number of

Kanbans can be reduced. However, if a company cannot afford

such alterations in their operation and the utilization rate

is very high, then the number of Kanbans required will in all

likelihood render the JIT technique ineffective.

An Analysis of Factors Influencing the Number of Kanbans Required at a Workcenter 102

4.4 SUMMARY

The first part of this chapter identified those factors

in the production environment that influence the number of

Kanbans that will be required at workcenters if a JIT system

is implemented. Four major factors were identified via an

analysis of the Japanese procedure for determining the number

of Kanbans required at a workcenter: throughput velocity,

the coefficient of variation in processing times, the machine

utilization rate and the autocorrelation between processing

times. For the Amer.ican production manager contemplating the

possibility of implementing the JIT technique in a production

operation that does not display the characteristics of a

Japanese operation, these factors can have a significant im-

pact on the effectiveness of the JIT technique.

The second part of the chapter described a simulation ap-

proach for determining workcenter lead times, and hence, the

number of Kanbans required at a workcenter necessary to pre-

vent backorders. This approach was based on the simplifying

(but logical) assumption that workcenters could be decoupled

and modeled separately as queueing systems in order to de-

termine lead times. A simulation analysis of an example

production operation indicated the procedure was effective

in determining the minimum number of Kanbans required at a

workcenter. This same simulation model was subsequently used

An Analysis of Factors Influencing the Number of Kanbans Required at a Workcenter 103

to test the validity of the factors identified in the first

part of the chapter as influencing the number of Kanbans re-

quired.

In this chapter it has been assumed that shop conditions,

while perhaps bad, are at least static. In the next chapter

a methodology for dynamically adjusting the number of Kanbans

in a shop in which conditions are not static is presented.

An Analysis of Factors Influencing the Number of Kanbans Required at a Workcenter 104

5.0 DYNAMICALLY ADJUSTING THE NUMBER OF KANSANS USING

ESTIMATED VALUES OF LEADTIME

Typically in a JIT operation the master production sched-

ule is frozen for one month and the number of Kahbans at each

workcenter is set based on the average demand for the period

[ 36]. When the monthly demand changes, one would expect that

the total number of Kanbans per month would also be changed.

Such is not the case at Toyota, as Yasuhiro Monden [36, p.

174] explains: "For example, suppose it is expected that the

average daily demand of next month will be two times the de-

mand of the current month. . . . At Toyota ... the total number

of Kanbans ... [is] unchanged." Companies using JIT such as

Toyota do not have to routinely adjust the number of Kanbans

from month to month for at least three reasons: they have a

large market share and hence demand variations from the

forecasted value are a small percentage of the total; they

have cross-trained workers whom they are able to switch from

workcenter to workcenter to mitigate temp_orary bottlenecks;

and their JIT shops. are so well run that they can handle

day-to-day problems as well as variations in demand. But

many firms either using or thinking of using Kanbans do not

exhibit these characteristics. They do not have large market

shares; workers do not have the training nor (with unions)

Dynamically Adjusting the Number of Kanbans Using Estimated Values of Leadtime 105

the opportunity to be switched from workcenter to workcenter

as needed; and these firms' JIT operations are still in the

lower portion of the learning curve, not to mention being

finely tuned and honed. In such firms it is often essential

to adjust the number of Kanbans.

Specifically, this chapter explores a workable method for

dynamically adjusting the number of Kanbans in a JIT shop.

After examining some basic principles of JIT with Kanbans,

the methodology is presented. This is followed by three ex-

amples. The purpose of the first example is to illustrate

in detail the methodology, while the last two examples pres-

ent solutions to interesting JIT problems using the dynamic

adjustment of Kanbans as presented in the methodology.

The Toyota Motor Company sets the number of Kanbans for

an i tern at a workcenter for a period using the following

formula [ 36]:

n = [ DL ( l + _a.) ] , (5-1)

where n is the number of Kanbans,

D is the average demand expressed in containers

L is the leadtirne for the i tern, which equals processing

time + waiting time + conveyance time + Kanban col-

lecting time [36, p.70]

a. is a safety factor which is set by the shop supervisor

to handle stochastic variation and anomalies

Dynamically Adjusting the Number of Kanbans Using Estimated Values of Leadtirne 106

and [x] means the smallest integer greater than or equal

to x.

This formula (if the safety factor is neglected) indicates

that the number of Kanbans depends on demand and the leadtime

necessary to produce a container's worth of goods, i.e., the

number of Kanbans is the smallest number of containers needed

to satisfy demand during lead time. If considered with the

idea of minimizing n, this equation summarizes the Japanese

philosophy of carrying only enough inventory to meet demand

during leadtime and of reducing leadtime as much as possible.

As mentioned, Toyota does not routinely adjust the number

of Kanbans once they are set because of its large market I

share, cross-trained workers, and well-run shops. With ref-

erence to equation (5-1), Toyota does not have to adjust n

because its large market share results in D generally being

kept within limits, and L is under control (i.e., does not

vary widely from its mean) because Toyota's workers are well

trained and the shops are well und.erstood and run. Further-

more, even if the product of D and L increases so that more

Kanbans are suggested at a workcenter, Toyota still does not

increase n. Instead, it takes other measures such as reas-

signing enough cross-trained workers to that workcenter to

reduce L by the amount required to hold n constant.

An important practical question is what should a company

do that cannot operate as Toyota does? The purpose of this

Dynamically Adjusting the Number of Kanbans Using Estimated Values of Leadtime 107

chapter is to answer this question and determine the condi-

tions under which a company should forego JIT with Kanbans

rather than dynamically adjusting n.

Al though equation ( 5-1) states that n may be determined

as the product of only two factors, (exogenous) demand and

(endogenous) leadtime, it should not be assumed that dynam-

ically adjusting the number of Kanbans is easy. Leadtime is

a function of many shop parameters, including utilization,

variability in processing times, and throughput rate of jobs

at the workcenter. Thus, leadtimes could not be simply

forecast statistically· using several years of leadtime data

because shop parameters themselves change with time. The

methodology in the next section explains how the most recent

set of shop conditions is used in conjunction with a forecast

of demand to set the number of Kanbans for the upcoming pe-

riod.

5. 1 METHODOLOGY FOR DYNAMICALLY ADJUSTING KANSANS

The basic principle underlying the methodology for the

dynamic adjustment of the number of Kanbans at a workcenter

is to exploit equation (5-1). Periodically (for example,

once a month), the number of Kanbans at the workcenter is

adjusted based on forecasted demand for the next month and

collected observations of leadtime at the workcenter during

Dynamically Adjusting the Number of Kanbans Using Estimated Values of Leadtime 108

the past month. In particular, the probability density

function (pdf) of leadtime is estimated and is combined with

the forecasted demand value to produce the pdf for n, the

number of Kanbans. Based on relative values of shortage and

holding costs at the workcenter, the supervisor may determine

the preferred number of Kanbans to be used the next period.

It was determined through experimentation that leadtime

observations are not independent but in fact are strongly

positively correlated. Thus it is necessary to also estimate

the autocorrelation function of leadtimes so that this factor

may be removed when estimating the leadtime pdf.

A "timeline" of our methodology is shown in Figure 5-1.

The methodology consists of two measuring periods, one to

estimate the autocorrelation function and the other to esti-

mate the pdf. These periods are followed by an "action,"

i.e., a possible adjustment in the number df Kanbans, and

finally a "settle" period which allows the shop to settle

down. before a new set of observations is made for the next

period.

Next, the detailed steps of the methodology are presented

for an item at a workcenter.

Step 0. Startup

If this shop is just being converted to Kanban operation

or major changes have been made which have perturbed shop

Dynamically Adjusting the Number of Kanbans Using Estimated Values of Leadtirne 109

0 ..

Initial Startup

Meas. Per. I

@ Meas. Settle Per. 2 Period

Meas. Per. I

@ Meas. Settle . Per. 2 Period

Meas. Per. I ...

I --------t-------J '------.. --l,i------...J•------ ... '

Cycle Period 2

Demond Period 2 I Cycle Period I

Demond• Period I 1 Demond Period 3

Figure 5-1. A "Timeline" Indicating the Different Activity Periods in the Methodology.

...

conditions, these transient effects should be permitted to

die out before step 1 is attempted.

Step 1. Measuring Period 1

The purpose of measuring period 1 is to obtain

autocorrelation estimates of the container leadtime series

so that statistically independent observations of leadtime

may be obtained during the second measuring period. How in-

dependence is achieved is explained in step 2.

Box and Jenkins [3, p. 33] suggest that "in practice, to

obtain a useful estimate of the autocorrelation function, we

would need at least fifty observations and the estimated

autocorrelations rk [at lag k] would be calculated for k = 0, 1, ... , K where K was not larger than say N/4." Since we

have observed significant autocorrelations at lags 20 - 25

(under admittedly severe conditions), we recommend that 100

observations be collected during measuring period l, if fea-

sible, but certainly no fewer than 50.

Once the (100) observations have been collected, the es-

timated autocorrelation at lag k may be computed using

Dynamically Adjusting the Number of Kanbans Using Estimated Values of Leadtime

------1'

111

N .l. I ( xt - x ) 2 N t=l

I k - 1,2, ... ,25, (5-2)

where x. is the jth observation of container leadtime. J

A correlogram may be drawn, if desired, to show the esti-

mated autocorrelation function.

After equation (5-2) has been calculated for all k listed,

the leadtime observations (from measuring period 1) should

be discarded.

Step 2. Measuring Period 2

The purpose of measuring period 2 is to estimate the pdf

of leadtimes for an item at the workcenter using the results

of step 1 to obtain statistically independent observations .

. Since statistical independence implies no correlation, the

correlogram of rk' s produced via equation ( 5-2) is examined

to determine the lag k beyond which all autocorrelations are

zero; in practice, the lag k is determined beyond which the

autocorrelation rernains below 0. 05. Then if observations

spaced k apart from each other are collected, the observa-

tions will be approximately statistically independent. Thus

step 2 is conducted to collect observations spaced k apart

and estimate the pdf of container lead times.

Dynamically Adjusting the Number of Kanbans Using Estimated Values of Ieadtime

'1 - - - ----------

112

The histogram historically has been used as an estimate

of pdf's, but Law and Kelton [29, pp. 180-181) point out that

more "modern" methods exist for this purpose. We recommend

that pdf' s be estimated using the 11 conventional 11 histogram

procedure, but recognize that sophisticated users of the

methodology will be able to generate better estimates using

other methods. In particular, a method developed by Tarter

and Kronmal [61) utilizing Fourier series estimates and the

orthogonal functions,

i

has been shown to overcome some disadvantages of the

histogram as a pdf estimator. Tarter and Kronmal' s method

has been computerized and the algorithm is described by

Tarter, et al. [60). They furnish an example based on 100

independent and identically distributed observations and

state that, based on their experience, "this demonstration

is representative, i.e., the nonparametric estimates obtained

are typical of what can be expected from samples of this

kind. II

Law and Kelton state in discussing the histogram procedure

that the choice of intervals and interval width is a "vexing"

part of the construction and that various widths should be

attempted until the histogram "looks smooth." The difficulty

Dynamically Adjusting the Number of Kanbans Using Estimated Values of Leadtime

----~---~------- ---- ------ ----- ---------T- - ------ --------------------------- -------------

113

that they allude to in this part of the process will be

somewhat alleviated by conditions described in step 4, and

will be discussed further at that point. For now note that

approximately 100 observations of Kanban leadtime should be

collected spaced k leadtimes apart and a histogram should be

developed from these observations. The histogram will be the

estimate of the pdf of L.

Step 3. Forecast Demand

Using standard (company) forecasting procedures, deter-

mine an estimate of the next demand period's demand, D, for

the item at the workcenter.

Step 4. Determine the pdf for the Number.of Kanbans

The purpose of this step is to estimate a probability

density function for n given the estimated pdf of leadtime

from step 2 and the forecast of demand for the next period

from step 3. To accomplish this the pdf of n' is determined,

where n' = DL, and then the pdf of n (where n [DL]) is

found.

f , (n') is defined as the pdf of the random variable n' n

and fL(L) as the pdf of the random variable L. Since D, once

estimated, is considered a deterministic constant over the

forecasted period, and since it is assumed that Dt-l will not

Dynamically Adjusting the Number of Kanbans Using Estimated Values of Leadtime

- --------- ---- ----- --- -------------------------y-- --- ----- -- ------- --- ------ -- ------. ----- --

114

differ 2 from Dt so drastically as · to _appreciably change

fL(L), it can be ~hown that (see, e.g., [44]),

1 L f I ( n I ) = - f (-) / d "f; 0 n ~ L ~ LJ D

(5-3)

Equation (5-3) states that n' has the same general pdf as

L, and is in fact just a scaled-down, reshaped version of

For example if 'D- = 2 and L fol.lows the exponential

density shown in Figure. 5-2a, then n' has the exponential pdf ,,,....

shown in Figure 5-2b. If D still equals 2 and L has the em-

pirically determined pdf shown in Figure 5-2d, f ,(n') is as n

shown in Figure 5-2e.

It remains to determine fn(n) given fn,(n'). The former

is just a discretized version of the latter with mass located

at n = 1,2,3, ... , and, density at each point k equal to k Jk-l fnr(n')dn,'. That is, the density at each discrete

Kanban value will be the area under the f , (n') curve between n the next lower number of Kanbans,and this number of Kanbans.

For example, the density at n = 3 Kanbans will be the area

under the f ,(n') curve between 2 and 3 Kanbans, because any n

For JIT to work, demand must be fairly constant. Minor fluctuations in demand are handled by adjusting the 1 ength of the workday. Thus, even if the dai 1 y demand fluctuates, the hourly demand rate should stay constant.

Dynamically Adjusting the Number of Kanbans Using Estimated Values of Leadtime 115

--' --' m

t .. • -_,

-

1.0

0.8

o.2t

1.0

0.9

0.8

0.1

I

_, -:. 0.6 -- 0.5

o_ ..

0.3-

0.2

.~. I z 3 4 5

L

(dt

m:••

·~ lo.II 0.1 ........ ~~

' I

05 I 1.5 2 2.5 J L

Co I 1.0 . s 0.5-• • r! 0.6.-

• -... c -j 0.2-I • 10

• - 0.5~. 'C-0.4· -.. ~ O.l

0.2·

(bl

c -c -O.'I

0.2

2 l .. 5 IO n'

C•I

o•l I0.!11

- 04·· (a't c

c 03 -02-

01 ·

123'156 n'

(cl

to.,9t

loa11.0&1 .•.

2 l 4·5 n

, ..

Co.s1

(O.ll

Co.on comn 23456

n Fi<Jtll"e 5-2. POF's of I., n', n for Two Illustrative Cases.

product of D and L greater than 2 and less than or equal to

3 will require 3 Kanbans according to equation (5-1). The

discretized version of Figure 5-2b is shown in Figure 5-2c,

and that of Figure 5-2e is shown in Figure 5-2f.

It was mentioned in step 2 that .all boundaries to con-

struct the histogram estimating f L ( L) would be condi ti oped

on results discussed in this step ( 5-4). Since fn(n) is a

discrete distribution taking on values only at the non-

negative integers, the cell boundaries in estimating fL(L)

should be set so that when f , (n') is constructed, no cell n . contains any integer as an interior point. This may be ac-

A complished by prohibiting cell? for fL(L) from containing j/D

as an interior point, where j = l, 2, For example, if A D = 10, then the points L = 0.1, 0.2, 0.3, ... should not be

inside any cell, but rather should be at cell boundaries.

Step 5. Determine the Minimum-Cost Number of Kanbans

In this step the density of n (fn(n)) is utilized to de-

termine the number of Kanbans to use for this i tern at this

workcenter for the next cycle period. To illustrate the

procedure, consider the hTI>othetical pdf fn(n) given in Fig-

ure 5-3. This density states that according to our best es-

timates, 40% of the time during the next demand period we

will need 1 Kanban at this workcenter, another 40% of the

time we will need 2, and the :test of the time (20%) we will

Dynamically Adjusting the Number of Kanbans Using Estimated Values of Leadtime 117

-c -;.4 .... ..,

• 4:.

(.4) (.4)

2 n

(.2)

Figure 5-3. A Hypothetical PDF fn(n) Used to Illustrate Step 5.

118

need 3 Kanbans. Since we don't know when during the next

cycle period we will need these differing number of Kanbans,

we select a single Kanban value and implement it f.or the en-

tire period.

In general there are two costs we may incur if we imple-

rrtent a single Kanban value when we have a pdf whose random

variable takes on more than one value (e.g., n = l, 2, or 3

as in Figure 5-3): a holding cost and a shortage cost. For

example, if we implement 2 Kanbans for this i tern at this

workcenter next period, then 40% of the time we will have one

Kanban too many and 20% of the time we will have one too few.

If we implement 1 Kanban, we will be 1 Kanban short 40% of

the time and 2 Kanbans short 20% of the time, but we will not

ever hold too many Kanbans. Likewise, if we implement 3

Kanbans, we will be 2 Kanbans "over" 40% of the time and also ' .1 Kanban over 40% of the time, but we will never be short.

If we denote as the cost of a shortage at a

workcenter/container/unit time and ch as the holding cost at

a workcenter/container/uni t time, then total cost computa-

tions may be made for each possible number of Kanbans. These

calculations are summarized in Table 5-1.

The breakeven point between 1 and 2 Kanbans occurs (see

Table 5-1) when

0.8cs = 0.4 ch + 0.2 c s

Dynamically Adjusting the Number of Kanbans Using Estimated Values of Leadtime 119

N 0

!!

4

5

6

Table 5-1. Cost Calculations Using the PDF for the Number of Kanbans of Figure 5-3.

Holding Cost Shortage Cost Total Cost

Och 0.4(1)c5 • 0.2(2)c1 Oen • o.acs

0.4(1)ch 0.2(1)c1 0.4ch • 0 2c . s

0.4(2)ch • 0.4(1)ch 0c, 1.2ch • Oc I

or when cs/ch = 2/3, while the break point between 2 and 3

Kanbans occurs when

0.4 ch + 0.2 c s

or when cs/ch = 4.

5-3 implies

Thus minimum cost for the pdf of Figure

Condition Optimal Number of Kanbans

1

2

3.

Step 5 may be stated as follows: Estimate cs and ch and de-

termine the number of Kanbans which minimizes the sum of

holding and shortage costs given the pdf for n determined in

step 4.

Note that if cs/ch-+ oo then step 5 implies that the optimal

number of Kanbans is the largest value the pdf takes on (3,

for the example), since shortages are so costly that none

should ever be incurred. Similarly, if cs/ch -+ 0, then the

optimal number of Kanbans is the minimum value the pdf real-

Dynamically Adjusting the Number of Kanbans Using Estimated Values of Leadtime 121

izes (1, in the example). Since in most JIT shops the cost

of a shortage significantly exceeds that of holding at most

non-final stage workcentera, we will discuss in detail the

special case cs/ch -+ 00 shortly.

Step 6. Act~on Step

Set the number of Kanbans for the item at this workcenter to

the value. determined in step 5. Note that if the number of

Kanba:ns is to be increased, then the additional containers

of goods will have to be produced in the upcoming period

during what would have been idle (i.e., non-processing) time.

Step?. Settle Period

The purpose of this step is to ensure that the workcenter

has sufficient time to settle down after step 6 has been im-

plemented and before step 1 is repeated. If the number of

Kanban.s is reduced by step 6, then there is no transient

settle period as the excess Kanban cards are merely pulled

off processed cbntainers and not recirculated. However, if

the number of Kanbans is increased by step 6, then there will

be a transient period of expected length,

E(length)

Dynamically Adjusting the Number of Kanbans Using Estimated Values of Leadtime

---· - -- --~---- - -- -

122

h + . w ere !:J. ni is the increase in Kanbans for item i at this

workcenter, util is the utilization of the workcenter, and

PT. is the average container processing time for item i. (The l

index i ranges over all items at the workcenter.)

Return to step 1 (i.e., the methodology repeats).

5.1.1 SPECIAL CASE OF THE METHODOLOGY: SHORTAGE COSTS

OVERWHELM HOLDING COSTS

It is frequently the case that cs/ch ~ ~ or at least that

cs/ch > M, where M is the largest breakeven point determined

from f (n) as in step 5. n The ratio cs/ch is often large in

a JIT shop because when a shortage of processed goods occurs

at a workcenter, it can delay work at workcenters located

toward the finished product as well as hold up workcenters

that furnish materials for this workcenter. (Recall that in

a JIT system preceding workcenters cannot make materials for

a workcenter unti 1 authorized to do so. If there is a

shortage at a workcenter, authorizaton at preceding

workcenters may be delayed until the shortage is removed and

there is a need to replace pre-process goods).

When shortage costs are considerably greater than holding

costs the above methodology may be greatly simplified. Note

that if f 1 (L) has finite support, i.e., is bounded by b above

and below by a (even if a and bare unknown), then,

Dynamically Adjusting the Number of Kanbans Using Estimated Values of Leadtime 123

/'. fn(n) = DfL (Lmax>·

max This phenomenon was described in passing while discussing

step 5 and results from the fact that shortages are so ex-

pensive that the maximum reasonable number of Kanbans should

be chosen.

It is known (see, for example, [311) that the pdf of the

maximum of a sample of size mis related to the depsity f and

the cumulative distribution function (cdf) F from which the

sample is taken as follows:

fx(m) (x) = fxmax(x) = m(Fx(x) )m-lfx(x) (5-4)

and that the cdf's are related by:

F ( )(x) = (F (x))m x m x (5-5)

Here the subscript x(m) denotes the m-th order statistic.

Note that if F has finite support on some interval [a, b],

an increase in m in equations (5-4) and (5-5) moves more and

more of the probability mass

toward the point b.

of f . (x) in equation (5-4) xmax

Since any "real-world" distribution of leadtime will be

bounded below by zero and above by some finite, perhaps un-

known number b, fL(L) has finite support. And since m 0 100

observations are taken during measuring period 2, the density

function of L will be a single-valued random variable with max all its mass (p = 1.0) at the point 'L , which is the maxi-max

Dynamically Adjusting the Number of Kanbans Using Estimated Values of Leadtime 124

mum of the collected leadtimes. 'L should be at (or very max close to) the point b. Thus

/'-fL (L) 6 ( L - Lmax) 0 6(L - b),

max

where 6(x - a) denotes an impulse of area one at the point x

= a.

As a result

This equation says that the density function of the number

of Kanbans consists of a single mass at the value n = [D

Thus to determine the number of Kanbans, one merely

needs to forecast D and to find the maximum leadtime over the

measurement period. For example, if D = 5 and L = 0.32, max then the number of Kanbans is

,,,.... ,...... n = [D Lmax] = [(5)(0.32)] = [1.6] = 2.

Due to the above simplifications the methodology for the

special case becomes:

Step 0. Startup - same as before.

Step 1. Measuring period 1 - same as before.

Step 2. Measuring period 2 - collect observations as before.

Now rather than estimating fL(L), estimate L , and use the max largest L obtained as this estimate. (Alternatively, a ser-

vice level may be specified. For example, if a 95% service

level is specified and 100 observations are taken, L0 _95 , the

Dynamically Adjusting the Number of Kanbans Using Estimated Values of Leadtime 125

95th percentile of fL(L), may be estimated using L( 9S)' the

95th order statistic. In this manner one is stipulating that

shortages projected for 5% of the time at the workcenter are

acceptable, as discussed earlier.)

Step 3. Forecast Demand - same as before.

Step 4. Determine the pdf for the Number of Kanbans - omit.

Step 5. Determine the Minimum-Cost Number of Kanbans - this

now becomes the simple calculation,

/' "" n = [D L ] . max (5-6)

Step 6. Action Step - same as before.

Step 7. Settle Period - same as before.

5.2 CASE EXAMPLES

Three examples will be presented in the remainder of the

chapter. The first is furnished to illustrate the details

of the methodology. The last two are used to demonstrate

that the methodology works well and are also of interest in

their own right.

The three examples are furnished in the context of the

example shop described in Chapter 2. The container process-

ing times and setup times for each product are shown in Table

5-2. In-process item costs are shown in Table 5-3. I

The

holding cost is 25% of container cost per year, the backorder

cost is $5.63 per container per hour, and, the setup cost is

Dynamically Adjusting the Number of Kanbans Using Estimated Values of Leadtime 126

Table 5-2. Container Processing Times and Shop Times for the Example Shop.

Container Processing Workcenter Time {hrs.~ Setup Number x y . Time -1 .0936 .0468 .0036 2 .0871 .0436 .0033 3 .2180 . 1090 .0088 4 .1453 .0727 .0055 s .3526 NIA NIA 6 NIA .3350- NIA

127

Table 5-3. Item Costs for the Example Shop.

!!!!!! Unit Cost

A S150

.e 150

c 150

0 150

E 412.S

F 37.S H. 37.S

300

J 450

K 37.S

L 37.S

x . 937.S y 787.S

128

$30 per hour. The shop is assumed to operate 8 hours per

shift, 2 shifts per day, 5 days per week. Demand is 19

containers/shift for final product X and 20 conta~ners/shift

for final product Y. The coefficient of variation (CV) of

processing times is 0.4 at each workcenter in the shop. With

such a large CV it is assumed that the shop is relatively new

to JIT and the workers are not yet totally accustomed to

procedures; hence the large. variances in processing times

relative to means. Using these parameters the example shop

runs at approximately 84% utilization.

5.2.1 EXAMPLE 1: ILLUSTRATION OF THE METHODOLOGY

The example shop was simulated at 84% utilization (as

previously indicated), and also, with processing times in-

creased to yield a 91% utilization case as well. The CV was

0.4 in both cases. It should be noted that the latter case

represents a shop pushed close "to the limit"; it has a high

utilization rate in addition to a very high variance in

processing times.

The methodology is now illustrated step by step using

workcenter 6 (which makes final product Y) as an example.

Other items at other workcenters are similar.

Dynamically Adjusting the Number of Kanbans Using Estimated Values of Leadtime 129

Step 0. Startup - Through trial and error it was determined

that a 1-week startup period was more than sufficient for

transient effects to die out.

Step 1. Measuring period 1 - Over 100 observations (in fact

400 over 2 weeks) were utilized to generate correlograms us-

ing equation ( 5-2) . These correlograms are shown in Figure

5-4a for the 84% utilization case and Figure 5-4b for 91%

utilization. !

The correlograms are decidedly different and

both show appreciable autocorrelations at large lags. (This

autocorrelation behavior is due to the large CV in the shop

which causes bursts of . long leadtimes followed later by

bursts of short leadtimes).

Step 2. Measuring period 2 - Utilizing the data that gener-

ated the correlograms shown in Figures 5-4a and 5-4b, it can

be observed that beyond lag 10 no autocorrelation exceeds

0. 05 for the 84% utilization case, and the same is true be-

yond lag 19 for the 91% case. Thus in measuring period 2,

11 independent 11 observations spaced every 10 leadtimes (i.e. ,

2 per shift) were collected for the lower utilization case,

and observations were spaced every 19 (~ctually rounded to

20) leadtimes (1 per shift) for the other case.

One hundred "independent" observations were collected for

each· case, which took 25 days in the former case and 50 in

Dynamically Adjusting the Number of Kanbans Using Estimated Values of Leadtime 130

-~1.0 - .9 ~ .e i:: .7 ~ _J .6 LU 0:: .~

~ .4

-g :: ~.I

I 2 3 4 ' 6 7 8 9 10 lag

Fiqure S-4a. Autocorrelation of Leadtime at a Workcenter (84' util.) .

-~ 1.0 - .9 z .8 0 - .1 ~ _J .6 LU .~ a:: a:: .4 0

~ .3 .a

~ . I

I a 3 4 s 6 7 a 9 10 II 12 13 14 I~ 16 17 18 19 20

lag

Figure S-4b. Autocorrelation of Leadtime at Workcente~ 6 (9ll util.)

131

the latter. The resulting pdf's for fL(L) are shown in Fig~

ures 5-5a and 5-5b. Note that increasing utilization dra-

matically changes fL(L). (For the reader's information, a

third case with the shop at 84% utilization and CV = 0.2 was

analyzed. This case showed that independent leadtimes could

be collected using every 3rd observation, thus greatly

shortening the length of measuring period 2.)

Step 3. Forecast Demand - Since demand is constant for final

product Y at workcenter 6, demand for the next demand cycle

period will be 20 containers/shift.

Step 4. Determine the pdf for the Number of Kanbans - The

density function for n' is obtained via equation (5-3) as,

Thus fn' (n') is scaled down vertically (by a factor of 20)

and enlarged horizontally (also by a factor of 20) from

The densities f , (n') are shown in Figures 5-6a and n

The discrete densities f (n) are shown in Figures 5-7a n

and 5-7b.

Step 5. Determine the Minimum-Cost Number of Kanbans - If

c = 1 and c = 10, then the total cost calculations for the h s

Dynamically Adjusting the Number of Kanbans Using Estimated Values of Leadtime 132

J

10• {9.Sl {9.8)

- Si-..J

6--..I _.,. 4 ..

2-(0.4l

0.0~ 0.10

LEAD TIME (hours)

a) 84% Utilization, CV•0.4

10

- e ..J -..I 6 _.,. 4

2

•

- (S.Sl ....

- <~:!!-.. (3.61 .. 0.0~ 0.10

0.1~

( 1.0l

0.1~

LEAD TIME (hours)

b) 91 % Utilization, CV•0.4

(1.0) l I

0.20 o.~

Figure 5-5. Histogram Estimates of fL{L) at Workcenter 6.

133

0.5 ... (0.49) . (0.49) .

0.4• -:s 0.3• -= .... 0.2-QI •

(0.02} ol---....JL---....J====i..--....... ---n' I 2 3 ' LEAD TIME CEMANO

a) 84~ UtiUUtion, CV-0.4

o.s :- 0.4 = ~ 0.3 .... a1

~.

... .. ·--..

(0.44)

(0.ZSl.

CO.IS)

{0;05) (0.0SJ 0.1 0 I I • n'

z 3 4 5 I . EAO TIME CEMAN[)

b) si % Utflfzation, CV=0.4

Figure 5-6. The PDF' s of Lead time Oernand, f n, (n') , at Workcenter 6.

--------

134

__, w Ul

O.!

- 0.4 c:: -0-3 c: -Q.2

0.1

(0.49l (Q49)

(.02) o----..__._....___....___...._....._._

2 3 4 5

NUMBER OF KANSANS

a} 84% Utf llzaUon, CV-0.4

-c:: -c: -o.s (.44)· 0.4

0.3 (.28)

0.2 (.18)

0.1 cos• to~n

0 2 3 4 5

NUMBER OF KANSANS

b) 91% Utilization, CV-0.4

Fiqure 5-7. The POF's of Number of Kanbans at Workcenter 6.

91% utilization case for n = 1, 2, 3, 4, and 5 Kanbans are

as shown in Table 5-4. The minimum-cost number of Kanbans

to be used for the next period is 4 for the given values of

ch and cs.

Step 6. Action Step - Implement 4 Kanbans at workcenter 6

for the next demand period for the 91% case.

Step 7. Settle Period - If there were 3 Kanbans at workcenter

6 during data collection, the expected length of the settle

period would be,

E(length) an +(PT ) = y = (4 - 3)(0.335 hours) = 3.72 hours

(1-util) 0.09

It would be reasonable to let the shop settle for one

shift, and possibly two, to be safe.

The methodology now returns to step 1 (measuring period

1) with 4 Kanbans circulating at workcenter 6.

5.2.2 EXAMPLE 2: ADJUSTING TO KANBAN MISSPECIFICATIONS

This example explores how well the methodology adjusts to a

one-time misspecification in the number of Kanbans. The in-

correct number of Kanbans could have been circulated at a

Dynamically Adjusting the Number of Kanbans Using Estimated Values of Leadtime 136

n

1 0

2 .28CU (U

3 .28(21 m

4 .28(3) Ill

5 .28(4) (1)

Table 5-4. Mini.mum Cost Calculations at Workcenter 6 .for the 91\ Utilization Case Assuming ch=l and c 5 =10.

Holding Cost Sboct.aqe Ct>s't.

.44CU UOI + .11 m ooa + .osm uoa + .o5C4t uoa

.180» (10) + . 05 (2) (10) + .05(3) UOI

+ .44UHU .o5m 1101 + .05(21 tlO)

+ .44(21 (11 +.18(11(1) • 05 (1) (10)

+ .44(3) UI .... 18 (2) (1) + .05111 Ill 0

Total Cost

11.50

4.38

2.50

2.40*

2.85

workcenter, for example, due to a badly missed forecast, or

because of a lack of knowledge on how to initialize a system

with Kanbans. This latter case not only happens when a shop

"opens its doors" for the first time, but also can arise af-

ter changes occur in the shop, such as with new or altered

products, different routings, etc.

Using previously developed techniques (by the author) it.

can be determined that the initial total number of Kanbans

needed in the sample shop is approximately 35, i.e., about 6

per workcenter on average. However, this initial number of

Kanbans was deliberately misspecified and the shop was loaded

with 72 Kanbans for this example.

As indicated in Figure 5-8, the shop ran for one period

at 72 Kanbans while data were being collected. The method-

ology predicted that 34 Kanbans should be used in the shop

the second period and that number was implemented. The num-

ber of Kanbans utilized during the rest of the two-year sim-

ulation oscillated between 34 and 38 Kanbans. (This

oscillation was caused by differences in leadtime estimates

created by variation in processing times from period to pe-

riod).

This example demonstrates that the shop can adjust very

quickly to incorrect Kanban numbers. A further implication

for shop supervisors and management is that one need not be

overly concerned with getting the number of Kanbans exactly

Dynamically Adjusting the Number of Kanbans Using Estimated Values of Leadtime 138

70

60

en .... 50 UJ en

z < cc z < 40 ~

u.. 0 0:: UJ 30 cc ~ z

20

10

2 4 5 s 1 a 9

TIME (months)

Figure 5-8. Example Illustrating the Effects of Misspecifying Initial Conditions.

139

-1 \II

"right" during startup or after major changes if one starts

with plenty of them. The methodology will adjust to proper

levels quickly.

5.2.3 EXAMPLE 3: A TRAINING CASE EXAMPLE

In this example the cost effects of training workers are

studied. In the first scenario, there is no training at all,

and the shop is run under the conditions described above.

This is the baseline case. In the second scenario, workers

are trained relatively slowly. The training is two-pronged

in its focus: the first aspect is to teach workers to reduce

setup times, and the second is to develop in them the ability

to meet processing time specifications more consistently.

The net effect of this gradual training is that setup times

decrease linearly over two years to one-half their original

values (which were given in Table 5-2), and, the coefficient

of variation (CV) of processing times also decreases linearly

over the two years from 0.4 to 0.2. In the third scenario,

workers are trained under a one-year concentrated program.

The effects are the same as in the second scenario, except

that the results take one year instead of two. Thus, CV and

setup times reach their terminal values at the end of year

1. They do not change further during the second year.

Dynamically Adjusting the Number of Kanbans Using Estimated Values of Leadtime 140

Cost results for the three scenarios of worker training

are shown in Table 5-5. Note that in this table the cost of

the worker training itself is not included. Excluding this

cost, "fast" training provides approximately $50, 000 in

savings over the no-training case. The savings come both

from reductions in setup costs and inventory costs. Once the

workers are trained these savings should accrue indefinitely.

The slow training case does provide about $35,000 improvement

over the no-training case, but saves less money than the

fast-training scenario.

For the interested reader the methodology functioned as

follows for the three training cases. In the fast-training

case the number of Kanbans was reduced to 24 by the end of

the first year, while it took two years for the same result

in the slower-training case. The no-training scenario had a

"steady-state" value of approximately 35 Kanbans.

5.3 SUMMARY

This chapter has demonstrated how manufacturers that do

not enjoy the luxury of cross-trained workers, large market

shares, and well-run and understood JIT shops can still

function in a Kanban environment. A methodology was devel-

oped for dynamically adjusting the number of Kanbans that

Dynamically Adjusting the Number of Kanbans Using Estimated Values of Leadtime 141

Table 5-5. Cost Results for the Three Scenarios of the Worker-Training Example.

No Slow Fast Cost Training Training Training

Inventory 200,700 173,987 162.224 Setup 31, 754 23,803 19,804

Backorder 0 0 0

Total 232,454 197,790 182,028

utilized forecasted demand and estimates of container

leadtime probability density functions.

Questions of actual implementation of the methodology,

such as should all measuring periods be synchronized across

the entire shop, were not addressed. It was deemed that they

were more properly resolved within the actual constraints of

a given shop.

The methodology is "safe" to use when sample sizes recom-

mended at each step are used. PDF estimates are not as

critical in this methodology as they might be in other ap-

plication areas because of the integer-nature of n. Each

leadtime observation, for example, needs to be placed in one

of a few, wide cells.

Some poorly run shops or other shops with severe internal

perturbations will find that . so many observations must be

collected, that they can only safely adjust the number of

Kanbans once every several months. If demand or conditions

in the shop change more rapidly than this, these shops should

not implement Just-in-Time. Just-in-Time is obviously not a

system for everyone. However, as was illustrated in the

final two examples, Kanbans can be dynamically adjusted in

shops in a sound and successful manner. For the firm still

learning about Just-in-Time, or the firm not possessing all

the cultural and technical luxuries of a Toyota, this is a

potentially significant result.

Dynamically Adjusting the Number of Kanbans Using Estimated Values of Leadtime 143

Chapter 6 concludes this study by summarizing the research

performed in light of the overall objectives of this disser-

tation and by suggesting further research endeavors regarding

JIT systems.

Dynamically Adjusting the Number of Kanbans Using Estimated Values of Leadtime 144

6.0 SUMMARY AND CONCLUSIONS

This research has investigated the feasibility of operat-

ing a Just-in-Time with Kanbans system within a sub-optimal

production environment typical of the environment faced by

many American manufacturing concerns. Analytical approaches

were developed for dealing with some of the problems caused

by this sub-optimal environment. These approaches were

tested using a Q-GERT simulation model of a hypothetical shop

with both job and' assembly operations employing a Just-in-

Time with Kanbans system.

The first part of this investigation examined whether a

JIT with Kanbans system should be attempted in an environment

where setup times cannot be reduced to levels that allow for

single container lotsizes at all workstations. There are

three important results from this section:

1. An American company which cannot reduce the setup times

at all workstations can still utilize JIT with Kanbans

by using Signal Kanbans at problem workstations.

2. A lotsizing technique using mathematical programming has

been demonstrated.

Summary and Conclusions 145

3. Inventory and setup costs must be weighed against each

other rather than simply seeking to reduce inventory to

its minimum level. Further, under certain conditions, a

Signal Kanban system at a workstation can be. more cost-

effective than a feasible standard Kanban system.

The second part of this investigation studied both the

factors present in the typical American production environ- _

ment that influence the number of Kanbans at a workstation

and how the initial number of Kanbans throughout a shop

should be determined. This section of the research makes

three important contributions.

1. A descriptive model of the relationship between the num-

ber of Kanbans required and the variability in processing

time, autocorrelation in processing times, workstation

utilization and throughput velocity is derived and sue-

cessfully verified using the example shop.

2. A method for determining the initial number of Kanbans

is demonstrated and successfully tested using the example

shop.

3. It is demonstrated that JIT with Kanbans can be used in

a manufacturing environment where the machine utilization

rate is high and there is a high degree of variability

Summary and Conclusions 146

and autocorrelation in processing times if a company is

willing to carry enough inventory. The cost of carrying

this large amount of inventory may, however, cause the

JIT with Kanbans system to be unprofitable or to be more

costly than alternative systems such as MRP.

The first two parts of the investigation when taken together

suggest that JIT with Kanbans can be used even if the shop

is operating under severe internal conditions such as long

setup times, high machine utilization rates, and processing

times with a high degree of variability and autocorrelation.

However, it has been assumed thus far that even though shop

conditions may be less than desirable for JIT operation, they

remain stable as do demand rates for final products.

The third part of the investigation examined dynamically

adjusting the number of Kanbans for the case where demand

and shop conditions are not stable. There are two important

results from this section:

1. A workable methodology for dynamically adjusting the

number of Kanbans

example shop.

is presented and demonstrated in the

2. The ability of the shop to adjust ~o changes corresponds

directly to how well the shop is functioning. If shop

conditions are good, i • e • / variability and

Summary and Conclusions 147

autocorrelation in processing times and utilization rates

are low, the shop can be adjusted every several weeks.

However, if the shop conditions are poor (i.e., the above

factors are at high levels), the adjustment cycle may

take several months.

The results from the three parts of the investigation when

taken together suggest that in an environment where demand

and shop conditions·are fairly stable, JIT with Kanbans can

still work despite high setup times, high variability in

processing tim~s, \~.utocorrelation in processing times, and

high utilization rates - conditions that one would expect to

find in many American firms considering the implementation

of JIT with Kanbans . These conditions can cause ·a signif-

icant increase in the amount of inventory required, but this

amount can be determined using the methodology which has been

presented. This conclusion in no way implies that JIT with

Kanbans should always be used, or that it is more cost-

effective than say MRP; it merely says that Kanbans is fea-

sible. Operating under these less than desirable conditions,

however, will in .all likelihood limit the ability of the shop

to adjust to an unstable environment such as when the firm

is unable to level the master production schedule, achieve

consistent performance from its workforce due to high em-

ployee turnover or labor strife, and so forth.

Summary and Conclusions 148

Monden suggests [34] that for the JIT with Kanbans system

to be successfully applied, the shop conditions must be "un-

der control" and demand must be relatively stable - condi-

tions which Toyota enjoys. The results from this

dissertation suggest that JIT with Kanbans is physically

feasible over a considerably broader region than Monden says

Toyota uses. "Feasible" here means not only that JIT has po-

tential to work, but rather that it will work and the level

of inventory carried can be computed using the methodology

developed.

The implications of this suggestion can be further delin-

eated using Figure 6~1. In that figure the horizontal axis

corresponds to how good shop conditions are, where by "shop

condi tio.ns" is meant the level of machine utilization, the

amount of variability and autocorrelation in processing

times, and the magnitude of throughput velocity. Recall that

in the second part of this investigation, these factors were

combined analytically into a Z -score: a high Z -score indi-p p

cates that shop conditions are "good", while a low z - score p

suggests that conditions are "poor.~'- The vertical axis in

Figure 6-1 represents the stability of demand and shop con-

ditions. Note that this axis reflects the rapidity with which

demand and shop conditions are changing, rather than the ab-

solute level of these factors. Monden suggests that the com-

pany considering use of JIT with Kanbans operate within the

small box-shaped area in the figure. The results presented

Summary and Conclusions 149

POOR

STABILITY

GOOD

MONDEN REGION

GOOD POOR

SHOP CONDITION

FIGURE 6-1. Feasible Region for JIT with Kanbans.

here indicate that Kanban is feasible and inventory levels

are predictable for the entire region under the curve - - a

much larger region then the Monden (Toyota) region.

The implication of this finding is twofold. First, a firm

can employ JIT with Kanbans even if its demand and shop con-

ditions are somewhat unstable, as long as shop conditions are

good in the sense of high Z -scores. For example, a shop with p high Z -scores at its workstations can tolerate master pro-p

duction schedules that cannot be frozen for as long as de-

sired. Such a shop would be able to adjust fairly rapidly to

these demand changes (using the methodology presented in the

third part of this research) since the adjustment cycle can

be relatively short due to the high Z -scores. Second, a firm p

can use JIT with Kanbans even if the manufacturing involves

"poor" shop conditions (in the sense of a low Z -score) as p

long as demand and shop conditions are stable. Such a firm

would require a long adjustment cycle in dynamically changing

the number of Kanbans, but this would not be a problem since

demand and shop conditions' stability do not require frequent

adjustments. However, if shop conditions are poor and insta-

bility is present, JIT with with Kanbans is not advisable

because frequent adjustments and long adjustment cycles are

required by the methodology. It is impossible to take a long

time to adjust and adjust frequently. Hence, the region

above the curve in Figure 6-1 is inappropriate for JIT en-

deavors.

Summary and Conclusions 151

The enlarged JIT with Kanban feasible region is poten-

tially good news to firms considering a change to JIT with

Kanbans They can make the change to JIT with Kanbans now

and accrue cost savings as they improve shop conditions and

stabilize demand (or as they get closer to the origin in

Figure 6-1).

6.1 EXTENSIONS AND FUTURE RESEARCH

These results point the way for future research in Just-

in-Time with Kanbans systems. First, while this research has

investigated the feasibility of using Kanbans under varying

conditions and the setting of inventory levels to handle

these conditions, it has not examined in detail the profit-

ability of doing so. It is more than likely that the costs

of operating in some areas of the feasible region will be

financially prohibitive. Additional research is needed to

determine in which portions of Figure 6-1 JIT with Kanbans

does have significant advantages over other sustems such as

MRP. Studies that have been performed comparing JIT to MRP

have taken the whole JIT with Kanban system and tried to ap-

ply it blindly in an American environment. The focus of this

research has been to adapt JIT to work in an American envi-

ronment. One path for future research is to compare MRP with

a JIT system that incorporates the results from this study.

Summary and Conclusions 152

A second area of research, very pragmatic in nature is to

validate this research on actual physical systems. In par-

ticular, it would be informative to install and then examine

an actual implementation of dynamic Kanban adjustment. "Real

world" measurements of processing time autocorrelations and

adjustment cycle lengths could be m~de, thereby establishing

the use of the methodology as is, or demonstrating the need

for more robust probability density function estimation.

Summary and Conclusions 153

\

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The vita has been removed from the scanned document